Proceedings of · The 4th Seminar on Harmonic Analysis and Applications Department of Mathematics,...

The 4th Seminar on Harmonic Analysis and Applications

Department of Mathematics, Kharazmi University, Tehran, Iran

January 20–21, 2016

Proceedings of

the 4th Seminar on

Harmonic Analysis and Applications

Kharazmi University, Tehran, Iran

20-21 January 2016



January 20–21, 2016

In the Name of God

Preface

On behalf of the Scientific Committee ofthe 4th Seminar on Harmonic Analysis and Applications

which was hold in Kharazmi University, 20-21 January 2016, I would like to thankall participants. This seminar was sponsored by the Ministry of Sciences, Researchand Technology, the Academy of Sciences and the Iranian Mathematical Society.

This book consists 50 papers presented in the seminar. We express our thanks to theauthors of the papers for their effort in the preparation of excellent contributions.

I would like to thank the members of the Scientific Committee as well as some othercolleagues who helped us in reviewing the papers carefully. In particular, I am gratefulto Dr A. H. Sanatpour who helped me in the preparation of this book.

Morteza EssmailiChairman of the Scientific Committee



January 20–21, 2016

Contents

Founder Committee . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii

Scientific Committee . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

IMS Representative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

Executive Committee . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

Titles of Papers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi–x

Papers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1–235

i



January 20–21, 2016

Founder Committee ofSeminar on Harmonic Analysis and Applications

Masoud Amini . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tarbiat Modares University

Hamid Reza Ebrahimi Vishki . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Ferdowsi University

Gholamhossein Esslamzadeh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shiraz University

Taher Ghasemi Honary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kharazmi University

Alireza Hosseiniuoun . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shahid Beheshti University

Rajabali Kamyabi Gol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ferdowsi University

Mahmoud Lashkarizadeh Bami . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Isfahan University

Alireza Medghalchi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kharazmi University

Rasoul Nasr-Isfahani . . . . . . . . . . . . . . . . . . . . . . . . . . . Isfahan University of Technology

Abdolrasoul Pourabbas . . . . . . . . . . . . . . . . . . . . . . Amirkabir University of Technology

Mohammad Ali Pourabdollah . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ferdowsi University

Mehdi Radjabalipour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shahid Bahonar University

Ali Rejali . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Isfahan University

Abdolhamid Riazi . . . . . . . . . . . . . . . . . . . . . . . . . . . Amirkabir University of Technology

ii



January 20–21, 2016

Scientific Committee

Masoud Amini . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tarbiat Modares University

Hamid Reza Ebrahimi Vishki . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Ferdowsi University

Golamhosein Esslamzadeh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shiraz University

Morteza Essmaili . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kharazmi University

Ali Ghaffari . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Semnan University


Azin Golbaharan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kharazmi University

Hossein Javanshiri . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yazd University

Rajabali Kamyabi Gol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ferdowsi University

Amir Khosravi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kharazmi University

Javad Laali . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Kharazmi University

Hakimeh Mahyar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kharazmi University

Alireza Medghalchi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kharazmi University

Rasoul Nasr-Isfahani . . . . . . . . . . . . . . . . . . . . . . . . . . . Isfahan University of Technology

Abdolrasoul Pourabbas . . . . . . . . . . . . . . . . . . . . . . Amirkabir University of Technology

Gholamreza Zabandan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kharazmi University

Amir H. Sanatpour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kharazmi University

iii



January 20–21, 2016

IMS Representative


iv



January 20–21, 2016

Executive Committee

Mohammad Ali Sobhanallahi Alireza Medghalchi

Seyed Hossein Serajzadeh Taher Ghasemi Honary

Azizollah Habibi Hakimeh Mahyar

Esmail Babolian Gholamreza Zabandan

Gholamreza Karimi Azin Golbaharan

Hossein Mohammadi Mohammad Reza Ghanei

Mehdi Godasiaee Morteza Essmaili

Hossein Ghavidel Amir H. Sanatpour

v



January 20–21, 2016

Titles of Papers

Closed ideals, point derivations and weak amenability of extended little LipschitzalgebrasD. Alimohammadi∗ and M. Mayghani . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1–5

Involutions and trivolutions on Banach algebrasA. Alinejad∗ and A. Ghaffari . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6–10

On frames for countably generated Hilbert modules over locally C∗-algebrasL. Alizadeh∗ and M. Hassani . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11–15

C∗-algebras and ordered Bratteli diagramsM. Amini, G. A. Elliott and N. Golestani∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16–20

Stability of approximate dual fusion frames in Hilbert spacesF. Arabyani Neyshaburi∗ and A. Arefijamaal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21–25

On stability of generalized dual Banach framesM. S. Asgari∗ and F. Enayati . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26–30

Generalized Riesz-dual sequences in Hilbert spacesM. S. Asgari and F. Enayati∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31–35

Harmonic analysis meets Lie theory via shearlet groupV. Atayi∗ and R. Kamyabi-Gol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36–38

Pseudoframe multiresolution structure on the space MN×N(C)-module L2(R,CN)H. Azarmi∗, M. Janfada and R. A. Kamyabi-Gol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39–43

Amenability and the compact multipliersA. Bagheri Salec∗ and M. Akbari Tootkaboni . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44–47

vi



January 20–21, 2016

Some basic theorems in operator valued measure theoryG. A. Bagheri-Bardi and M. Khosheghbal-Ghorabayi∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48–52

The Stone-Cech compactification of groupoidsF. Behrouzi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53–58

Separating ideals in the algebra of uniformly continuous integrable functionsM. Dashti∗ and R. Nasr-Isfahani . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59–63

Module mean for Banach algebrasH. Ebrahimi∗ and A. Bodaghi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64–71

Operator algebras of weighted conditional expectation operatorsY. Estaremi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .72–76

Daws conjecture on the Arens regularity of B(X)R. Faal∗ and H. R. Ebrahimi Vishki . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77–79

Projectivity of c0(S , ω−1)E. Feizi and J. Soleymani∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80–84

Homological properties of some class of Fréchet algebrasE. Feizi∗ and J. Soleymani . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85–89

The amenability of Banach algebrasK. Haghnejad Azar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .90–94

Multi-bounded sets and projectivity of certain Banach modulesF. Hamidi Dastjerdi and S. Soltani Renani∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95–99

The BSE property of certain Banach algebrasZ. Kamali . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100–104

On the weak∗ continuity of LUC(G)∗-module action on LUC(X,G)∗ related toG-space XH. Javanshiri∗ and N. Tavallaei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105–108

vii



January 20–21, 2016

The second dual of certain triple systemsA. A. Khosravi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109–111

Frames, operators and duality principleA. Khosravi and F. Takhteh∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112–116

On the converse of a theorem due to B. Forrest, E. Kaniuth, A. T. M. Lau and N. SpronkJ. Laali and M. Fozouni∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117–120

∗-Fusion frames in Hilbert modules over locally C∗-algebrasT. Lal Shateri . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121–124

Dual Banach algebras; a type of φ-contractibilityA. Mahmoodi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125–129

Trivolutions in algebras related to second duals of hypergroup algebrasA. R. Medghalchi and R. Ramazani∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130–133

Ternary n-weak amenability of C∗-algebras and group algebrasM. R. Miri and M. Niazi∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134–138

On local information functional for discrete dynamical systemsU. Mohammadi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139–143

On some solvable extensions of the Heisenberg groupM. Nasehi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144–148

On the BSE-property of abstract Segal algebrasM. Nemati . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149–153

Module extension Banach algebras and (σ, τ)-amenability of Banach algebrasA. Niazi Motlagh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154–160

Derivation on Hilbert H∗-modulesM. Niknam∗ and E. Keyhani . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161–165

viii



January 20–21, 2016

Crossed product of triple operator spacesH. Nikpey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166–168

Structure of some Beurling algebras on left coset spacesB. Olfatian Gillan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .169–172

Topics in the concept of amenability modulo an ideal of Banach algebrasH. Rahimi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173–176

On n-weak amenability of Lau product of Banach algebrasM. Ramezanpour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177–179

Some Banach algebra properties in A ×T BN. Razi∗ and A. Pourabbas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180–186

Convolutions on the Haagerup tensor products of Fourier algebrasM. Rostami∗ and N. Spronk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187–192

Wavelets and amenability of a locally compact groupJ. Saadatmandan∗ and A. Bagheri Salec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193–197

Arens regularity of bounded bilinear maps and weakly compactnessA. Sahleh and L. Najarpisheh∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198–200

An applications of Fourier transform in the study of projections of probabilitymeasuresE. Salavati . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201–205

Fusion frame multipliersM. Shamsabadi∗ and A. A. Arefijamaal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206–210

Some properties of weighted composition operators on L2(Σ)S. Shamsi Gamchi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211–214

On the cohomological properties of dual Banach algebrasA. Shirinkalam∗ and A. Pourabbas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215–219

ix



January 20–21, 2016

Approximate amenability of semigroup algebra modulo an idealA. Soltani∗ and H. Rahimi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220–223

The weighted KPC-hypergroupsS. M. Tabatabaie and F. Haghighifar∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224–228

A dense additive subsemigroup of L2(G)S. M. Tabatabaie . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .229–231

Paradoxical decomposition of hypergroupsA. Yousofzadeh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232–235

x



January 20–21, 2016

Closed Ideals, Point Derivations and Weak Amenability ofExtended Little Lipschitz Algebras

D. Alimohammadi∗ and M. Mayghani

Abstract

Let (X, d) be a compact metric space and let K be a nonempty compact subset of X. Let α ∈ (0, 1]and let Lip(X,K, dα) denote the Banach algebra of all continuous complex-valued functions f on Xfor which p(K,dα)( f ) = sup | f (x)− f (y)|

dα(x,y) : x, y ∈ K, x , y < ∞ when equipped with the algebra norm|| f ||Lip(X,K,dα) = || f ||X + p(K,dα)( f ), where || f ||X = sup| f (x)| : x ∈ X. We denote by lip(X,K, dα) the closedsubalgebra of Lip(X,K, dα) consisting of all f ∈ Lip(X,K, dα) for which | f (x)− f (y)|

dα(x,y) → 0 as d(x, y) → 0with x, y ∈ K. In this paper we show that every proper closed ideal of (lip(X,K, dα), ∥ · ∥Lip(X,K,dα)) is theintersection of all maximal ideals containing it. We also prove that every continuous point derivation oflip(X,K, dα) is zero. Next we show that lip(X,K, dα) is weakly amenable if α ∈ (0, 1

2 ). We also prove thatlip(T,K, d

12 ) is weakly amenable where T = z ∈ C : |z| = 1, d is the Euclidean metric on T and K is a

nonempty compact set in (T, d).

2010 Mathematics subject classification: Primary 46J10; Secondary 46J15.Keywords and phrases: Banach function algebra, Extended Lipschitz algebra, Point derivation, Weakamenability.

1. Introduction

Let A be a complex algebra and let φ be a multiplicative linear functional on A. Alinear functional D on A is called a point derivation on A at φ if

D( f g) = φ( f )Dg + φ(g)D f ,

for all f , g ∈ A. A multiplicative linear functional φ on A is called a character on A ifφ( f ) , 0 for some f ∈ A. We denote by Car(A) the set of all characters on A which iscalled the carrier space of A. For each φ ∈ Car(A), we denote by ker(φ) the set of allf ∈ A for which φ( f ) = 0. Clearly, ker(φ) is a proper ideal of A.

Let (A, ∥ · ∥) be a commutative unital complex Banach algebra. We know thatCar(A) , ∅ and it is a compact Hausdorff space with the Gelfand topology. Moreover,ker(φ) is a maximal ideal of A for all φ ∈ Car(A) and every maximal ideal of A has theform ker(ψ) for some ψ ∈ Car(A). Let I be an ideal of A. The hull of I is the set of all

∗ speaker

1

D. Alimohammadi andM. Mayghani

φ ∈ Car(A) such that φ( f ) = 0 for all f ∈ I. We denote by hull(I) the hull of I. Let Sbe a nonempty subset of Car(A). We define ker(S ) :=

∩φ∈S

ker(φ),

IA(S ) = f ∈ A : there is an open set V in Car(A) with S ⊆ V

such that φ( f ) = 0 f or all φ ∈ V,

and JA(S ) = IA(S ), the closure of IA(S ) in (A, ∥ · ∥). Clearly, ker(S ) and JA(S ) areclosed ideals of A and S is contained in hull(JA(S )).

Let A be a complex Banach algebra and X be a Banach A-module. The set of allcontinuous X-derivations on A is a complex linear space, denoted by Z1(A,X). The setof all inner X-derivations on A is a complex linear subspace of Z1(A,X), denoted byB1(A,X). The quotient space Z1(A,X)

/B1(A,X) is denoted by H1(A,X) and called the

first cohomology group of A with coefficients in X. We say that A is weakly amenableif H1(A, A∗) = 0. This definition is first given by Johnson in [4]. It is known [2] thatif A is a commutative unital complex Banach algebra and A has a nonzero continuouspoint derivation, then A is not weakly amenable. The following result is known.

Theorem 1.1. Let A and B be complex Banach algebra, A be weakly amenable andΦ : A → B be a continuous algebra homomorphism such that Φ(A) is dense in B.Then B is weakly amenable.

Let X be a compact Hausdorff space. We denote by C(X) the commutative unitalcomplex Banach algebra consisting of all complex-valued continuous functions on Xunder the uniform norm on X which is defined by ∥ f ∥X = sup| f (x)| : x ∈ X.

A complex Banach function algebra on X is a complex subalgebra A of C(X) suchthat A separates the points of X, contains 1X (the constant function on X with value 1)and it is a unital Banach algebra under an algebra norm ∥ · ∥.

Let (A, ∥ · ∥) be a Banach function algebra on X. For each x ∈ X, the mapex : A → C, defined by ex( f ) = f (x) ( f ∈ A), is an element of Car(A) which is calledthe evaluation character on A at x. It follows that A is semisimple and ∥ f ∥X ≤ ∥ f ∥Car(A)

for all f ∈ A. Moreover, the map EX : X → Car(A) defined by EX(x) = ex is injectiveand continuous. If EX is surjective, then we say that A is natural. In this case, EX isa homeomorphism from X onto Car(A). It is known that if (A, ∥ · ∥) is a self-adjointinverse-closed Banach function algebra on X, then A is natural.

Let (A, ∥ · ∥) be a commutative unital complex Banach algebra. Then A is calledregular if for every proper closed subset S of Car(A) and each φ ∈ Car(A) \ S , thereexists an f in A such that f (φ) = 1 and f (S ) = 0, where f is the Gelfand transformof f .

Let A be a complex Banach function algebra on a compact Hausdorff X. If A isregular, then for each proper closed subset E of X and each x ∈ X \ E there exists afunction f in A such that f (x) = 1 and f (E) = 0. Moreover, the converse of theabove statement holds whenever A is natural.

2

Closed Ideals, Point Derivations and Weak Amenability of Extended...

Let (X, d) be a metric space. For α > 0, a complex-valued function f on X isa Lipschitz function of order α on X if there exists a positive constant M such that| f (x) − f (y)| ≤ M(d(x, y))α for all x, y ∈ X.

Let (X, d) be a compact metric space and α ∈ (0, 1]. We denote by Lip(X, dα) theset of all complex-valued Lipschitz function on (X, dα). Then Lip(X, dα) is a complexsubalgebra of C(X) and 1X ∈ Lip(X, dα). Moreover, Lip(X, d) separates the points ofX. For a nonempty subset K of X, and a complex-valued function f on K, we set

p(K,dα)( f ) = sup | f (x) − f (y)|dα(x, y)

: x, y ∈ K, x , y.

Clearly, f ∈ Lip(X, dα) if and only if p(X,dα)( f ) < ∞. The dα-Lipschitz norm ∥·∥Lip(X,dα)on Lip(X, dα) is defined by ∥ f ∥Lip(X,dα) = ∥ f ∥X+p(X,dα)( f ). Then (Lip(X, dα), ∥·∥Lip(X,dα))is a commutative unital complex Banach algebra. The set of all complex-valuedfunctions f on X for which lim

d(x,y)→0

| f (x)− f (y)|dα(x,y) = 0, is a closed complex subalgebra of

(Lip(X, dα), ∥ · ∥Lip(X,dα)) containing 1X . This algebra is called little Lipschitz algebraof order α on (X, d) and denoted by lip(X, dα). The structure of ideals and pointderivations of Lipschitz algebras studied by Sherbert in [5].

Bade, Curtis and Dales studied the weak amenability of little Lipschitz algebras in[2] and obtained the following results that we use them in the sequel.

Theorem 1.2 (see [2, Theorem 3.10]). Let (X, d) be a compact metric space andα ∈ (0, 1

2 ). Then lip(X, dα) is weakly amenable.

Theorem 1.3 (see [2, Theorem 3.13]). Let d be the Euclidean metric on T. Thenlip(T, d

12 ) is weakly amenable.

Let (X, d) be a compact metric space, K be a nonempty compact subset of Xand α ∈ (0, 1]. We denote by Lip(X,K, dα) the set of all f ∈ C(X) for whichp(K,dα)( f ) < ∞. Clearly, Lip(X, dα) ⊆ Lip(X,K, dα) and Lip(X,K, dα) = Lip(X, dα)if K = X. Moreover, Lip(X,K, dα) is a self-adjoint inverse-closed complex subalgebraof C(X). It is easy to see that Lip(X,K, dα) is a unital Banach algebra under the algebranorm ∥ · ∥Lip(X,K,dα) defined by

∥ f ∥Lip(X,K,dα) = ∥ f ∥X + p(K,dα)( f ) ( f ∈ Lip(X,K, dα)).

Therefore, (Lip(X,K, dα), ∥·∥Lip(X,K,dα)) is a natural Banach function algebra on X. Thisalgebra is called extended Lipschitz algebra of order α on (X, d) with respect to K. Wedenote by lip(X,K, dα) the set of all f ∈ C(X) for which lim

d(x,y)→0x,y∈K

| f (x)− f (y)|dα(x,y) = 0. Clearly,

lip(X,K, dα) is a complex subalgebra of Lip(X,K, dα) containing 1X and a closed set in(Lip(X,K, dα), ∥ · ∥Lip(X,K,dα)). This algebra is called extended little Lischitz algebra oforder α on (X, d) with respect to K. Clearly, Lip(X, d) ⊆ lip(X, dα) ⊆ lip(X,K, dα) andlip(X,K, dα) = lip(X, dα) for α ∈ (0, 1) if K = X. Moreover, lip(X,K, dα) is self-adjointinverse-closed and so (lip(X,K, dα), ∥ · ∥Lip(X,K,dα)) is also a natural Banach functionalgebra on X. It is clear that Lip(X,K, dα) = lip(X,K, dα) = C(X) whenever K is finite.

3

D. Alimohammadi andM. Mayghani

Extended Lipschitz algebras and extended little Lipschitz algebras were first intro-duced in [3]. The following result is obtained in [1] that we use it in the sequel.

Theorem 1.4 (see [1, Corollary 2.9]). Let (X,d) be a compact metric space, K bea nonempty compact subset of X and α ∈ (0, 1). Then lip(X, dα) is dense in(lip(X,K, dα), ∥ · ∥Lip(X,K,dα)).

In Section 2, we show that extended little Lipschitz algebras are regular. We alsodetermine the structure of closed ideals of these Banach algebras. In Section 3, weshow that every continuous point derivation on an extended little Lipschitz algebra iszero. In Section 4, we show that certain extended little Lipschitz algebras are weaklyamenable.

2. Closed Ideals of Extended Little Lipschitz Algebras

Throughout this section we always assume that (X, d) is a compact metric space, Kis a nonempty compact subset of X and α ∈ (0, 1).

Theorem 2.1. Let A = lip(X,K, dα). Then (A, ∥ · ∥Lip(X,K,dα)) is regular.

Lemma 2.2. Let f ∈ lip(X,K, dα) with Z( f ) , ∅ where Z( f ) = x ∈ X : f (x) = 0.Then there exists a sequence fn∞n=1 in lip(X,K, dα) satisfying:(i) for each n ∈ N, there is an open set Un in X with Z( f ) ⊆ Un such that

fn|Un = f |Un ,(ii) lim

n→∞∥ fn∥Lip(X,K,dα) = 0.

Theorem 2.3. Let A = lip(X,K, dα) and I be a closed ideal of (A, ∥ · ∥Lip(X,K,dα)). ThenI = ker(hull(I)).

Corollary 2.4. Let A = lip(X,K, dα). and I be a proper ideal of A. Then the followingstatements are equivalent.(i) I is a closed ideal of (A, ∥ · ∥Lip(X,K,dα)).(ii) There exists a closed subset Y of X such that I = ker(EX(Y)).(iii) I is the intersection of all maximal ideals of A containing it.

3. Point Derivations of Extended Little Lipschitz Algebras

Let (A, ∥ · ∥) be a commutative complex unital Banach algebra and I be an ideal

of A. Set I2 =

n∑

i=1αi figi : n ∈ N, αi ∈ C, fi, gi ∈ I (i ∈ 1, ..., n)

. Clearly, I2 is also

an ideal of A. For each φ ∈ Car(A), we denote by Dφ the set of all continuous pointderivations on A at φ.

Theorem 3.1. Let (A, ∥ · ∥) be a commutative complex unital Banach algebra with unit1 and φ ∈ Car(A). Then(i) D ∈ A∗ is a point derivation at φ if and only if D1 = 0 and D f = 0 for all

f ∈ (ker(φ))2.

4

Closed Ideals, Point Derivations and Weak Amenability of Extended...

(ii) Dφ \ 0 , ∅ if and only if ker(φ) , (ker(φ))2.

Theorem 3.2. Let (X, d) be a compact metric space, K be a nonempty compact subsetof X, α ∈ (0, 1) and A = lip(X,K, dα). If φ ∈ Car(A), then (ker(φ))2 = ker(φ).

Theorem 3.3. Let (X, d) be a compact metric space, K be a nonempty compact subsetof X, α ∈ (0, 1) and A = lip(X,K, dα). Then every continuous point derivation on(A, ∥ · ∥Lip(X,K,dα)) is zero.

4. Weak Amenability of Certain Extended Little Lipschitz Algebras

Let (X, d) be a compact metric space, K be a compact subset of X, α ∈ (0, 1) andA = lip(X,K, dα). By Theorem 3.3, A has some chance of being weakly amenable. Wegive some sufficient conditions that (A, ∥ · ∥Lip(X,K,dα)) to be weakly amenable.

Theorem 4.1. Let (X, d) be a compact metric space, K be a nonempty compact subsetof X, α ∈ (0, 1

2 ) and A = lip(X,K, dα). Then (A, ∥ · ∥Lip(X,K,dα)) is weakly amenable.

Theorem 4.2. Let d be the Euclidean metric on T, K be a nonempty compact set in(T, d) and α ∈ (0, 1

2 ]. Then (lip(T,K, dα), ∥ · ∥Lip(T,K,dα)) is weakly amenable.

[1] D. Alimohammadi and S. Moradi, Sufficient conditions for density in extended Lipschitz algebras,Caspian Journal of Mathematical Sciences 3 (1) (2014), 151-161.

[2] W. G. Bade, P. G. Curtis and H. G. Dales, Amenability and weak amenability for Beurling andLipschitz algebras, Proc. London Math. Soc. (3)35 (1987), 359-377.

[3] T. G. Honary and S. Moradi, On the maximal ideal space of extended analytic Lipschitz algebras,Quaestiones Mathematicae 30 (3)(2007), 349-353.

[4] B. E. Johnson, Derivations from L1(G) into L1(G) and L∞(G), Proc. International Confrenceon Harmonic Analysis, Luxamburg, (1987) (Lecture Notes in Math. Springer Verlag).

[5] D. R. Sherbert, The structure of ideals and point derivations in Banach algebras of Lipschitzfunctions, Trans. Amer. Math. Soc. 111(1964), 240-272.

D. Alimohammadi,Department of Mathematics,Faculty of Science,Arak University,Arak, 38156-8-8349, Irane-mail: [email protected]

M. Mayghani,Department of Mathematics,Payame Noor University,Tehran, 19395-3697, Irane-mail: [email protected]

5



January 20–21, 2016

Involutions and trivolutions on Banach algebras

A. Alinejad∗ and A. Ghaffari

Abstract

We investigate involutions and trivolutions on Banach algebras.We study positive linear functionals ona trivolutive Banach algebras. We also show that when G is an amenable group, the second dual of theFourier algebra of G admits an involution extending one of the natural involutions of A(G) if and only ifG is finite. However, A(G)∗∗ always admits trivolution.

2010 Mathematics subject classification: 46K05, 43A20, 43A30, 43A10, 46H05.Keywords and phrases: Arens multiplication, positive linear functional, involution, trivolution, Fourieralgebra.

1. Introduction and preliminaries

By a well-known result of Civin and Yood [3, Theorem 6.2], if A is a Banachalgebra with an involution θ : A −→ A, then the second (conjugate-linear) adjointθ∗∗ : A∗∗ −→ A∗∗ is an involution on A∗∗ (with respect to either of the Arens products)if and only if A is Arens regular; when this is the case, θ∗∗ is called the canonicalextension of θ. Grosser [11, Theorem 1] has shown that if A has a bounded right [left]approximate identity, then a necessary condition for existence of an involution on A∗∗

with respect to the first [second] Arens product is that A∗ has to be factorizable on theright [left], in the sense that A∗ · A = A∗ [A · A∗ = A∗].

Let G be a locally compact group, for the special case of the group algebra, L1(G),Grosser showed that for any infinite, non-discrete, locally compact group G, L1(G)∗∗

does not admit any algebra involution with respect to either of the Arens products[11, Theorem 2]. Farhadi and Ghahramani in [6] have shown that if G has an infiniteamenable subgroup, then the second dual algebra L1(G)∗∗ does not admit an involutionextending the natural involution of L1(G). (See also the related papers [7, 14, 16]).

Recently, Filali, Monfared, and Singh in [7] introduced and studied trivolutions onBanach algebras A as a continuous conjugate-linear, anti-homomorphism τ on A suchthat τ3 = τ. They obtained many characterizations of trivolutions. Trivolutions aremore availble than involutions [7].

∗ speaker

6

A. Alinejad and A. Ghaffari

We investigate existence of involutions and trivolutions on Banach algebras andthe second dual of the Furier algebras. An outline of the present paper is as follows.In section 2, we start with a general study of trivolutions on Banach algebras. Westudy positive linear functionals on a trivolutive Banach algebra and we investigaterelationship between the positive linear functionals on A and A♯. In Theorem (2.3)we show that, unlike C∗-algebras, Gelfand-Naimark-Segal theorem does not hold evenwhen the range of trivolution is a C∗-algebra. In section 3, we give a brief study ofinvolutions and trivolutions on the second dual of the Fourier algebra of G. We provethat when G is an amenable locally compact group, then A(G)∗∗ has an involution, withrespect to either of the Arens products, if and only if G is finite. However, the Banachalgebra A(G)∗∗ itself always admits trivolution. Moreover, when G is amenable, A(G)∗∗

admits a trivolution with range A(G) if and only if G is compact.We recall the definition of the Arens products on the second dual A∗∗ of a Banach

algebra A [1, 2]. For a, b ∈ A, f ∈ A∗, and F,G ∈ A∗∗, the first Arens product FG isdefined by

⟨ f · a, b⟩ = ⟨ f , ab⟩, ⟨G · f , a⟩ = ⟨G, f · a⟩, ⟨FG, f ⟩ = ⟨F,G · f ⟩.Similarly the second Arens product F^G is defined by the identities

⟨a · f , b⟩ = ⟨ f , ba⟩, ⟨ f · F, a⟩ = ⟨F, a · f ⟩, ⟨F^G, f ⟩ = ⟨G, f · F⟩.The Banach algebra A is said to be Arens regular whenever the two Arens productscoincide, i.e. FG = F^G, for all F,G ∈ A∗∗.

Let P(G) denote the subspace of Cb(G) consisting of all continuous positive definitefunctions on G, and let B(G) be its linear span. Then B(G) can be identified with thedual of the group C∗-algebra C∗(G), the completion of L1(G) under its largest C∗-norm.Also, B(G) with poitwise multiplication and the dual norm is a commutative Banachalgebra, the Fourier-Stieltjes algebra of G [5].

Let Pρ(G) be the closure of P(G) ∩ Cc(G) in the compact-open topology, andBρ(G) its linear span. Then Bρ(G) is a closed ideal in B(G), and is the dual of theC∗-algebra C∗ρ(G), where C∗ρ(G) is the norm closure of ρ( f ); f ∈ L1(G) in B(L2(G))and ρ( f )(h) = f ∗ h for h ∈ L2(G) [5]. As is known, Bρ(G) = B(G) if and only if G isamenable.

2. Trivolutions on Banach algebras

Positive elements in a trivolutive algebra were defined in [7]. Let (A, τ) be atrivolutive algebra and let x ∈ A. Then x is called positive if x is hermitian (i.e.τ(x) = x) and x = τ(y)y for some y ∈ A. We denote the set of all positive elements byA+. Let P be a linear functional on a trivolutive algebra (A, τ). Then P is a positivefunctional if P(x) ≥ 0 for all x ∈ A+.

It follows from the definition that if x is hermitian, then x ∈ A+ if and only ifx = yτ(y) for some y ∈ A. The latter condition holds if and only if x = τ2(y)τ(y) forsome y ∈ A (i.e. A+ = τ2(y)τ(y) : y ∈ A). So, P ≥ 0 if and only if P(τ2(y)τ(y)) ≥ 0for all y ∈ A. The last statement means that P ≥ 0 if and only if P

∣∣∣τ(A) ≥ 0.

7


Theorem 2.1. Let P be a positive linear functional on a trivolutive complex algebra(A, τ). Then for all x, y ∈ A,

(i) P(τ2(x)τ(y)) = P(τ2(y)τ(x)).(ii) |P(τ2(x)τ(y))|2 ≤ P(τ2(x)τ(x)) P(τ2(y)τ(y)).

Let us also recall from [7] that an element x in a unital trivolutive algebra (A, τ)with identity e is called unitary if xτ(x) = τ(x)x = e. We denote the set of all unitaryelements by Au.

Theorem 2.2. Let (A, τ) be a trivolutive normed algebra and A♯ be the unitized algebrawith the norm ∥(λ, x)∥ = |λ| + ∥x∥, and let τ♯ be a trivolution extension of τ on A♯. LetP be a positive linear functional on A. Then there is a positive linear functional P♯ onA♯ that agrees with P on A if and only if

(i) P(τ(x)) = P(τ2(x)),(ii) |P(τ(x))|2 ≤ αP(τ2(x)τ(x))for all x ∈ A, where α is a positive number.

Let (A, τ) be a trivolutive algebra and τ∗ be the conjugate-linear adjoint of τ definedby ⟨τ∗( f ), a⟩ = ⟨ f , τ(a)⟩, ( f ∈ A∗, a ∈ A). If f : A −→ C is a linear functional on A,then f τ := τ∗( f ), is also a linear functional on A. It is easily to check that the mapf −→ f τ, is conjugate-linear and in general f τττ = f τ. If (A, τ) is normed, then∥ f τ∥ ≤ ∥ f ∥. We call f hermitian if f τ = f [7]. Let P be a positive linear functional onthe trivolutive algebra A. Then for all a ∈ A,

Pτ(τ(a)τ2(a)) = P(τ(τ(a)τ2(a))) = P(τ3(a)τ2(a))

= P(τ(a)τ2(a)) = P(τ(a)τ2(a)) ≥ 0.

Thus, Pτ ≥ 0. In the same way the converse is true (i.e. P ≥ 0 if and only if, Pτ ≥ 0).It is well known that any positive linear functional on a C∗-algebra A is hermitian.

However, for a trivolutive Banach algebra (A, τ), this is not true. Indeed, let 0 , a ∈ Abe an element in the closed ideal I = ker τ. As an application of the Hahn-Banachtheorem, there is non-zero Pa ∈ A∗ annihilating B = τ(A) such that Pa(a) , 0. ThenPa is positive, however, it is not hermitian.

Theorem 2.3. Let (A, τ) be a trivolutive Banach algebra such that A is a stronglysplitting extension of a C*-algebra B. If P is a positive linear functional on A,then there is a cyclic representation (πP,HP) of A with unit cyclic vector ξ such thatPτ(a) = (πP(a)ξ|ξ) for all a in A.

Remark 2.4. Let H be an infinite-dimensional separable Hilbert space. It is wellknown that K(H), the compact operators on H , is the only nontrivial closed two-sided ideal of B(H) and by [4], K(H) is not complemented in B(H). Hence B(H)does not admit any trivolution except the usual involutions. Moreover, if (A, τ) is atrivolutive Banach algebra, by [7, theorem 2.5], any representation of A vanishes onker τ.

8

A. Alinejad and A. Ghaffari

3. involution and trivolution on second dual of fourier algebra

The Fourier algebra of G denoted by A(G) was introduced by Eymard in [5].Each φ ∈ A(G) has the form φ(x) = (λ(x)h|k) where h, k ∈ L2(G) and λ is theleft regular representation of G on L2(G). It is a closed ideal in B(G) [5]. AlsoA(G) ⊂ Bρ(G). Consider VN(G), the von Neumann subalgebra of B(L2(G)) generatedby the representation λ. Then A(G) is the predual of VN(G), and therefore, there is thenatural module action of A(G) on VN(G). When G is commutative, and G is the dualgroup of G, then L1(G) A(G) via the Fourier transform (for more details see [5]).Let UCB(G) be the subspace of VN(G) as the norm closure of A(G) · VN(G). ThenUCB(G) is a C∗-subalgebra of VN(G) and elements in UCB(G) are called uniformlycontinuous functionals on A(G). For more information one may consult [9, 10, 12, 13].

A linear functional m on VN(G) is called a topological invariant mean, if ∥m∥ =⟨m, I⟩ = 1 (where I = λ(e) is the identity operator) and if ⟨m, φ · T ⟩ = φ(e)⟨m,T ⟩ forall T ∈ VN(G), φ ∈ A(G). It can be shown that the set of topological invariant meanson VN(G) is always non-empty [15, Theorem 4]. We note that ⟨φ, I⟩ = φ(e) for allφ ∈ A(G). Finally, for φ ∈ B(G) we define

φ(x) = φ(x); φ(x) = φ(x−1), x ∈ G. (1)

It is easy to check that all of the above operations are involutions on B(G).

Theorem 3.1. Let G be an amenable locally compact topological group. Then A(G)∗∗

admits an involution extending any one of the involutions in (1) on A(G) if and only ifG is finite.

Theorem 3.2. Let G be a locally compact Hausdorff group. Then the followingstatements hold:(i) A(G)∗∗ admits a trivolution with range Bρ(G).(ii) The Banach algebra UCB(G)∗ has a trivolution with range Bρ(G).

Theorem 3.3. Let G be a compact group. Then A(G)∗∗ admits a trivolution with rangeA(G). When G is amenable, the converse holds.

[1] R. Arens, The adjoint of a bilinear operation, Proc. Amer. Math. Soc. 2 (1951), 839–848.[2] , Operations induced in function classes, Monatsh. Math. 55 (1951), 1–19.[3] P. Civin and B. Yood, The second conjugate space of a Banach algebra as an algebra, Pacific J.

Math. 11 (1961), 847–870.[4] J. B. Conway, The compact operators are not complemented in B(H), Proc. Amer. Math. Soc. 32

(1975), 549–550.[5] P. Eymard, L’algèbre de Fourier d’un groupe localement compact, Bull. Soc. Math. France 92

(1964), 181–236.[6] H. Farhadi and F. Ghahramani, Involutions on the second duals of group algebras and a multiplier

problem, Proc. Edinb. Math. Soc. 50 (2007), 153–161.[7] M. Filali, M. Sangani Monfared and A. I. Singh, Involutions and trivolutions in algebras related

to second duals of group algebras, Illinois Journal of Mathematics, Vol. 57, Number 3, (2013),755–773.

9


[8] E. E. Granirer, Weakly almost periodic and uniformly continuous functionals on the Fourieralgebra of any locally compact group, Trans. Amer. Math. Soc., 189, (1974), 371–382.

[9] , Density theorems for some linear subspaces and some C∗-subalgebra of VN(G), Sym-pos. Mat., vol. XXII, Istit. Nazionale di Alta Mat., (1977), 61–70.

[10] , On group representations whose C∗-algebra is an ideal in its von Neumann algebra,Ann. Inst. Fourier (Grenoble) 29 (1979), 37–52.

[11] M. Grosser, Algebra involutions on the bidual of a Banach algebra, Manuscripta Math. 48 (1984),291–295.

[12] A. T.-M. Lau, Uniformly continuous functionals on the Fourier algebra of any locally compactgroup, Trans. Amer. Math. Soc. 251 (1979), 39–59.

[13] , The second conjugate algebra of the Fourier algebra of a locally compact group, Trans.Amer. Math. Soc. 267 (1981), 53–63.

[14] M. Neufang, Solution to Farhadi–Ghahramani’s multiplier problem, Proc. Amer. Math. Soc. 138(2010), 553–555.

[15] P. F. Renauld, Invariant means on a class of von Neumann algebras, Trans. Amer. Math. Soc. 170(1972), 285–291.

[16] A. I. Singh, Involutions on the second duals of group algebras versus subamenable groups,Studia Math. 206 (2011), 51–62.

A. Alinejad,College of Farabi,University of Tehran,P.O. Box, 3718117469, Iran.e-mail: [email protected]

A. Ghaffari,Department of Mathematics,Semnan University,P.O. Box, 35195-363, Semnan, Iran.e-mail: [email protected]

10



January 20–21, 2016

On frames for countably generated Hilbert modules over locallyC∗-algebras

L. Alizadeh∗ and M. Hassani

AbstractLet X be a countably generated Hilbert module over a locally C∗-algebra A in multiplier module M(X)of X. We propose the necessary and sufficient condition such that a sequence hn : n ∈ N in M(X) isa standard frame of multipliers in X. We also show that if T in b(LA(X)), the space of bounded maps inLA(X), is surjective and hn : n ∈ N is a standard frame of multipliers in X, then T hn : n ∈ N is astandard frame of multipliers in X, too.

2010 Mathematics subject classification: Primary 46L08, Secondary 46L05, 42C15.Keywords and phrases: locally C∗-algebras, Hilbert modules over locally C∗-algebras, bounded modulemaps, countably generated Hilbert modules, standard frames of multipliers.

1. Introduction

Locally C∗-algebras are generalizations of C∗-algebras. Locally C∗-algebras were firstintroducted by A. Inoue in [5] and were also studied more by N. C. Phillips (under thename of pro-C∗-algebra) in [10].

A locally C∗-algebra is a complete Hausdorff complex topological ∗-algebra A,whose topology is determined by its continuous C∗-seminorms in the sense that thenet aii∈I converges to 0 in A if and only if the net p(ai)i∈I converges to 0 for everycontinuous C∗-seminorm p in set S (A) of all continuous C∗-seminorms onA.

Hilbert modules are essentially objects like Hilbert spaces by allowing the innerproduct to take values in a (locally) C∗-algebra rather than the field of complexnumbers. The notion of Hilbert module over locally C∗-algebras generalize the notionof Hilbert C∗-module. Hilbert modules over locally C∗-algebras were first consideredby N. C. Phillips in [10]. He showed that many properties of the Hilbert C∗-modulesare valid for Hilbert modules over locally C∗-algebras. But the main body of the workon Hilbert modules over locally C∗-algebras is due to M. Joita, all her work on thesubject can be found in her book, under the title "Hilbert modules over locally C∗-algebras" see [6].

Here we recall some results about Hilbert modules over locally C∗-algebras from[10] and [6].∗ speaker

11

L. Alizadeh, M. Hassani

A (right) pre-Hilbert module over a locally C∗-algebra A (or a pre-Hilbert A-module) is a complex vector space X which is also a right A-module, compatiblewith the complex algebra structure, equipped with an A-valued inner product ⟨·, ·⟩ :X×X −→ Awhich is C- andA-linear in its second variable and satisfies the followingrelations:1. ⟨y, x⟩ = ⟨x, y⟩∗ for every x, y ∈ X;2. ⟨x, x⟩ ≥ 0 for every x ∈ X;3. ⟨x, x⟩ = 0 if and only if x = 0.

A pre-Hilbert A-module X is a Hilbert A-module if X is complete with respectto the topology determined by the family of seminorms pXp∈S (A) where pX(x) =√

p(⟨x, x⟩), x ∈ X.If A is a locally C∗-algebra, then A is a Hilbert A-module with ⟨a, b⟩ = a∗b, and

the set HA of all sequences (an)n with an ∈ A such that Σna∗nan converges in A isa Hilbert A-module with the action of A on HA defined by (an)nb = (anb)n and theinner product defined by ⟨(an)n, (bn)n⟩ = Σna∗nbn.

Let A be a locally C∗-algebra and let X be a Hilbert A-module. A subset Y of Xis a generating set for X if the closed submodule of X generated by Y is the whole ofX. We say that X is countably generated if it has a countable generating set.

Let A be a locally C∗-algebra and let X and Y be Hilbert A-modules. An A-module map T : X −→ Y is called bounded if for all p ∈ S (A), there is Mp > 0such that pY(T x) ≤ Mp pX(x) for all x ∈ X, and it is adjointable if there is a mapT ∗ : Y −→ X such that ⟨T x, y⟩ = ⟨x,T ∗y⟩ for all x ∈ X, y ∈ Y. It is easy to seethat every adjointable map is a bounded A-module map. The set of all adjointablemaps from X into Y is denoted by LA(X,Y) and we write LA(X) for LA(X,X).The vector space LA(X,Y) is a complete locally convex space with respect to thetopology defined by the family of seminorms pX,Yp∈S (A), where pX,Y defined bypX,Y(T ) = sup pY(T x); x ∈ X and pX(x) ≤ 1. In particular, LA(X) becomes a locallyC∗-algebra with respect to the topology defined by the family of seminorms pXp∈S (A).

We say that an element T of LA(X,Y) is bounded in LA(X,Y) if there is M > 0such that pX,Y(T ) ≤ M for all p ∈ S (A). The set of all bounded elements inLA(X,Y) is denoted by b (LA(X,Y)). It is clear that the map ∥ · ∥∞ defined by∥T∥∞ = suppX,Y(T ); p ∈ S (A) is a norm on b (LA(X,Y)). And b (LA(X,Y)) isa Banach space with respect to the norm ∥ · ∥∞. So b (LA(X)) is a C∗-algebra withrespect to the norm ∥ · ∥∞.

Frames for Hilbert spaces were introduced by Duffin and Schaeffer [3] in 1952as part of their research in non-harmonic Fourier series. They were reintroduced anddeveloped in 1986 by Daubechies, Grossmann and Meyer [2]. Many generalizationsof frames were introduced, meanwhile, M. Frank and D. Larson presented a generalapproach to the frame theory in Hilbert C∗-modules in [4]. A frame for a countablygenerated Hilbert C∗-module X is a sequence xn : n ∈ N for which there areconstants C,D > 0 such that

C⟨x, x⟩ ≤∑

n

⟨x, xn⟩⟨xn, x⟩ ≤ D⟨x, x⟩, x ∈ X.

12

On frames for countably generated Hilbert modules over locally C∗-algebras

M. Joita generalized this definition to the situation Hilbert modules over locally C∗-algebras in [7]. A frame of multipliers for a countably generated Hilbert module Xover a locally C∗-algebra A is a sequence hn : n ∈ N in M(X) multiplier module ofX for which there are constants C,D > 0 such that

C⟨x, x⟩ ≤∑

n

⟨x, hn⟩⟨hn, x⟩ ≤ D⟨x, x⟩, x ∈ X. (1)

We consider standard frames of multipliers for which the sum in the middle of (1)converges inA for every x ∈ X.

Now we recall some facte about multiplier modules from [8] and [7].LetA be a locally C∗-algebra and let X be a HilbertA-module. It is not difficult to

check that LA(A,X) is a Hilbert LA(A)-module with the action of LA(A) on LA(A,X)defined by t.s = ts, t ∈ LA(A,X) and s ∈ LA(A) and the LA(A)-valued inner-productdefined by ⟨t, s⟩ = t∗ s. Moreover, pLA(A,X)(s) = pLA(A,X)(s) for all s ∈ LA(A,X) andfor all p ∈ S (A), the topology on LA(A,X) induced by the inner product coincideswith the topology determined by the family of seminorms pLA(A,X)p∈S (A). ThereforeLA(A,X) is a Hilbert LA(A)-module and since LA(A) can be identified with themultiplier algebra M(A) ofA (see [10] and [6]), LA(A,X) becomes a Hilbert M(A)-module. The Hilbert M(A)-module LA(A,X) is called the multiplier module of X,and it is denoted by M(X).

The map iX : X −→ M(X) defined by iX(x)(a) = xa, x ∈ X and a ∈ A embeds Xas a closed submodule of M(X). Moreover, if t ∈ M(X) then t.a = t(a) for all a ∈ Aand ⟨t, x⟩ = t∗(x) for all x ∈ X.

A Hilbert A-module X is countably generated in M(X) if there is a countable sethn; hn ∈ M(X), n = 1, 2, ... such that the closed submodule of M(X) generated byhn.a; a ∈ A, n = 1, 2, ... is the whole of X.

If X is a countably generated Hilbert A-module, then X is countably generatedin M(X). In general, X is not countably generated when X is countably generated inM(X).

Let X be a Hilbert A-module. A sequence hnn in M(X) is a standard frame ofmultipliers in X if for each x ∈ X,

∑n⟨x, hn⟩⟨hn, x⟩ converges in A, and there are two

positive constants C and D such that C⟨x, x⟩ ≤ ∑n⟨x, hn⟩⟨hn, x⟩ ≤ D⟨x, x⟩ for all x ∈ X.

The numbers C and D are called lower and upper frame bounds.Any countably generated HilbertA-module X in M(X) admits a standard frame of

multipliers.In this paper we extend some results from [1] in the contex of Hilbert modules over

locally C∗-algebras.

2. Main results

First, we investigate some properties of bounded A-linear maps between HilbertA-modules.

Definition 2.1. An element T of LA(X,Y) is bounded below in LA(X,Y) if there isM > 0 such that MpX(x) ≤ pY(T x) for all p ∈ S (A) and x ∈ X.

13

L. Alizadeh, M. Hassani

Our first result is a generalization of Proppsition 2.1 from [1].

Proposition 2.2. Let A be a locally C∗-algebra, X and Y Hilbert A-modules andT ∈ b(LA(X,Y)). The following statements are mutually equivalent:

1. T is surjective;2. T ∗ is bounded below in LA(Y,X), i.e., there is m > 0 such that mpY(x) ≤

pX(T ∗x) for all p ∈ S (A) and x ∈ Y.3. T ∗ is bounded below with respect to inner product, i.e., there is m′ > 0 such that

m′⟨x, x⟩ ≤ ⟨T ∗x,T ∗x⟩ for all x ∈ Y.

Corollary 2.3. Let A be a locally C∗-algebra, X a Hilbert A-module and T ∈b(LA(X)) such that T ∗ = T. The following statements are mutually equivalent:

1. T is surjective;2. There are m,M > 0 such that mpX(x) ≤ pX(T x) ≤ MpX(x) for all p ∈ S (A) and

x ∈ X.3. There are m′, M′ > 0 such that m′⟨x, x⟩ ≤ ⟨T x,T x⟩ ≤ M′⟨x, x⟩ for all x ∈ X.

Remark 2.4. Let X be a countably generated HilbertA-module in M(X) and let hnnbe a standard frame of multipliers in X. The module morphism θ : X −→ HA definedby θ(x) = (⟨hn, x⟩)n is called the frame transform for hnn. The frame transform θ is aninjective adjointable module morphism from X to HA with closed range. Moreover,θ ∈ b(LA(X,HA)) which realizes an embedding of X onto an orthogonal summand ofHA, The adjoint operator θ∗ is surjective. Moreover, θ∗ θ is an invertible element inb(LA(X)). For details we refer the reader to [7].

Theorem 2.5. Let A be a locally C∗-algebra, X a countably generated Hilbert A-module in M(X), hn : n ∈ N a sequence in M(X) and θ(x) = (⟨hn, x⟩)n∈N for x ∈ X.The following statements are mutually equivalent:

1. hn : n ∈ N is a standard frame of multipliers in X.2. θ ∈ b(LA(X,HA)) and θ is bounded below in LA(X,HA).3. θ ∈ b(LA(X,HA)) and θ∗ is surjective.

Another direct consequence of Proposition 2.2 is that if T ∈ b(LA(X)) is surjectiveand hn : n ∈ N is a standard frame of multipliers in X, then T hn : n ∈ N is astandard frame of multipliers in X, too.

Theorem 2.6. Let A be a locally C∗-algebra, X a countably generated Hilbert A-module in M(X), and T ∈ b(LA(X)) surjective. If hn : n ∈ N is a standard frame ofmultipliers in X with frame bounds C and D, then T hn : n ∈ N is a standard frameof multipliers in X with frame bounds C||(TT ∗)−1||−1

∞ and D||T ||2∞.

The next result show that the condition (1) from the definition of standard framescan be replaced with a weaker one.

Theorem 2.7. Let A be a locally C∗-algebra, X a countably generated Hilbert A-module in M(X), and hn : n ∈ N a sequence in M(X) such that

∑n⟨x, hn⟩⟨hn, x⟩

14

On frames for countably generated Hilbert modules over locally C∗-algebras

converges in A for every x ∈ X. Then hn : n ∈ N is a standard frame of multipliersin X if and only if there are constants C,D > 0 such that

C pX(x)2 ≤ p

∑n

⟨x, hn⟩⟨hn, x⟩ ≤ DpX(x)2, x ∈ X, p ∈ S (A). (2)

[1] L. Arambasic, On frames for countably generated Hilbert C∗-Modules, Proc. Amer. Math. Soc.,135(2007), 469-478.

[2] I. Daubechies, A. Grossman, Y. Meyer, Painless nonorthogonal expansions, J. Math. Phys.,27(1986), 1271-1283.

[3] R. J. Duffin, A. C. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc.,72(1952), 341-366.

[4] M. Frank, D. R. Larson, Frames in Hilbert C*-modules and C*-algebras, J. Operator Theory,48(2002), 273-314.

[5] A. Inoue, Locally C∗-algebra, Memoirs of the Faculty of Science, Kyushu University, Ser. A,25(1971), 197-235.

[6] M. Joita, Hilbert modules over locally C∗-algebras, Univesity of Bucharest Press, (2006), 150 pg.ISBN 973737128-3.

[7] M. Joita, On frames in Hilbert modules over pro-C∗-algebras, Topology and its applications,156(2008), 83-92.

[8] M. Joita, On multipliers of Hilbert modules over pro-C∗-algebras, Studia Math., 185(2008), 263-277.

[9] E. C. Lance, Hilbert C∗-modules. A toolkit for operator algebraists, London Mathematical SocietyLecture Note, Series 210, Cambridge University Press, Cambridge 1995.

[10] N. C. Philips, Inverse limits of C∗-algebras, J. Operator Theory, 19(1988), 159-195.

L. Alizadeh,Department of Mathematics,Mashhad Branch,Islamic Azad University,Mashhad, Irane-mail: [email protected]

M. Hassani,Department of Mathematics,Mashhad Branch,Islamic Azad University,Mashhad, Irane-mail: [email protected]

15



January 20–21, 2016

C*-Algebras and Ordered Bratteli Diagrams

M. Amini, G. A. Elliott and N. Golestani∗

Abstract

There are four categories which are closely related to each other: C*-algebras, Bratteli diagrams, orderedabelian groups, and dynamical systems. Some relations between these categories are naturally categorical(i.e., the morphisms are also considered). For instance, K-theory provides two functors K0 and K1 fromthe category of C*-algebras to the category of ordered abelian groups. However, most of these relationsare just at the level of objects. We construct suitable functors between these categories. In particular, wegive a functorial formulation of a result of Herman, Putnam, and Skau.

2010 Mathematics subject classification: Primary 46L05, Secondary 37B05.Keywords and phrases: C*-algebra, dynamical system, ordered Bratteli diagram, crossed product.

1. Introduction

There are four categories which are closely related to each other: C*-algebras, Brattelidiagrams, ordered abelian groups groups, and dynamical systems. There are standardprocedures to go from one to the other. For example, to each AF C*-algebra Ait is associated an infinite directed graph B(A) called its Bratteli diagram [4]. Toeach dynamical system (X, φ) (where X is a compact Hausdorff space and φ is ahomeomorphism of X) it is associated a C*-algebra crossed product C(X) ⋊φ Z (see[12]). Also, to each C*-algebra A it is associated two ordered abelian groups K0(A)and K1(A) (see [11]).

Some of the above-mentioned relations are naturally categorical (i.e., the mor-phisms are also considered). For instance, K-theory provides two functors K0 and K1from the category of C*-algebras to the category of ordered abelian groups. However,most of these relations are just at the level of objects (see for example [4], [8], and[7]). We have started to define suitable functors between these categories and obtainsome applications of this approach [1–3]. In this note—which is an abridged versionof some sections of [3]—we focus on the ordered Bratteli diagrams, Cantor minimalsystems, and crossed product C*-algebras.

In 1972, Bratteli introduced what are now called Bratteli diagrams and used themto study AF algebras [4]. He showed that the map B : AF → BD from the category∗ speaker

16

M. Amini, G. A. Elliott and N. Golestani

of AF algebras to the set of Bratteli diagrams has the property that for A1, A2 ∈ AF,A1 A2 if and only if B(A1) is equivalent to B(A2). In [1], the authors proposed acategory structure for Bratteli diagrams and showed that the map B : AF → BD is astrong classification functor. This completes the Bratteli’s work on the classificationof AF algebras. Let us recall the following definition.

Definition 1.1 (Elliott, [6]). A functor F : C → D is called a classification functor ifF(a) F(a) implies a b, for each a, b ∈ C; it is called a strong classification functorif each isomorphism from F(a) to F(b) is the image of an isomorphism from a to b.Thus in this case we have a b if and only if F(a) F(b).

In 1976, based on the Bratteli’s work, Elliott introduced the notion of a dimensiongroup (a certain ordered abelian group) and showed that the map K0 : AF → DGfrom the category of Bratteli diagrams to the category of dimension groups is a strongclassification functor. The classification of unital simple amenable separable C*-algebras using the K-theoretic invariants (Elliott’s program) is one of the main researchareas of the theory of operator algebras. In [1], we showed that the categories BDand DG are equivalent and the functors B and K0 are essentially the same from acategorical point of view.

In 1981, A. V. Versik used Bratteli diagrams to construct so-called adic transforma-tions [10]. Based on his work, Herman, Putnam, and Skau introduced the notion of anordered Bratteli diagram and associated a dynamical system to each (essentially sim-ple) ordered Bratteli diagram [8]. They showed that there is a one-to-one correspon-dence between essentially simple ordered Bratteli diagrams and essentially minimalsystems [8, Theorem 4.7]. This correspondence was used effectively to study Cantorminimal dynamical systems and characterization of various types of orbit equivalencein terms of isomorphism of related C*-crossed products and dimension groups [7, 8].

In Section 2 we define the morphisms between ordered Bratteli diagrams and weobtain the category of ordered Bratteli diagrams OBD. We show that the correspon-dence of Herman, Putnam, and Skau, mentioned above, is in fact an equivalence ofcategories. In Section 3 we construct the functorsA andAF .

2. Ordered Bratteli Diagrams

Definition 2.1 ([8], Definition 2.1). A Bratteli diagram consists of a vertex set V andan edge set E satisfying the following conditions. We have a decomposition of V as adisjoint union V0 ∪V1 ∪ · · · , where each Vn is finite and non-empty and V0 has exactlyone element, v0. Similarly, E decomposes as a disjoint union E1∪E2∪· · · , where eachEn is finite and non-empty. Moreover, we have maps r, s : E → V such that r(En) ⊆ Vnand s(En) ⊆ Vn−1, n = 1, 2, 3, . . . (r = range, s = source). We also assume that s−1vis non-empty for all v in V and r−1v is non-empty for all v in V \ V0.

We denote such a B by the diagram

V0E1 // V1

E2 // V2E3 // · · · .

17


Definition 2.2 ([8], Definition 2.3). An ordered Bratteli diagram is a Bratteli diagram(V, E) together with a partial order ≥ on E such that e and e′ are comparable if andonly if r(e) = r(e′). That is, we have a linear order on each set r−1v, v ∈ V \ V0.

Definition 2.3. Let B = (V, E,≥) and C = (W, S ,≥) be ordered Bratteli diagrams.An ordered premorphism f : B → C is a triple (F, ( fn)∞n=0,≥) where (F, ( fn)∞n=0) is apremorphism (see [1, 2]) and ≥ is a partial order on F such that:(1) e, e′ ∈ F are comparable if and only if r(e) = r(e′), and ≥ is a linear order on

r−1v, v ∈ W;(2) the diagram of f : B→ C commutes:

V0E1 //

F0

V1E2 //

F1

V2E3 //

F2||||||||

· · ·

W0 S 1

// W1 S 2

// W2 S 3

// · · · .

We define an equivalence relation on ordered premorphisms and we obtain orderedmorphisms (see [2] for details).

Theorem 2.4. The class OBD with ordered morphisms, as defined above, is a category.

Definition 2.5 (cf. [8], Definition 1.2). Let X be a compact metrizable space. Let φ bea homeomorphism of X and y ∈ X. The triple (X, φ, y) is called an essentially minimaldynamical system if the dynamical system (X, φ) has a unique minimal (non-empty,closed, invariant) set Y and y ∈ Y.

Denote by SDS the category whose objects are quadruples (X, φ, y,R) where(X, φ, y) is an essentially minimal dynamical system, X is totally disconnected, andR is a system of Kakutani-Rohlin partitions for (X, φ, y) (see [2]). The morphisms areas follows. Let (X, φ, y,R) and (Y, ψ, z,S) be in ODS. By a morphism α : (X, φ, y,R)→(Y, ψ, z,S) we mean a homomorphism from the dynamical system (X, φ) to (Y, ψ) (i.e.,a continuous map with α φ = ψ α) such that φ(y) = z. See [2] for the definition ofthe map P : SDS→ OBD.

Theorem 2.6. The map P : SDS→ OBD is a contravariant functor.

Denote by OBDes the full subcategory of OBD containing essentially simpleordered Bratteli diagrams. The following is the main result of this section. It is afunctorial formulation of [8, Theorem 4.7].

Theorem 2.7. The functor P : SDS→ OBDes is an equivalence of categories.

An application of the above theorem in dynamical systems is in the study offactors and extensions of a Cantor minimal system. In fact, the functor P providesa model (diagram) for each homomorphism between these dynamical systems and byTheorem 2.6 each homomorphism comes from this method.

18

M. Amini, G. A. Elliott and N. Golestani

3. Crossed Products Functors

In this section we construct the functors A and AF from the category of essentiallyminimal t.d. dynamical systems to the category of unital C*-algebras and AF algebras.

Definition 3.1. Define the category of essentially minimal totally disconnected dynam-ical systems DS as follows. The objects of this category are the essentially minimaltotally disconnected dynamical systems. Let (X, φ, y) and (Y, ψ, z) be in DS. By a mor-phism α : (X, φ, y) → (Y, ψ, z) in DS we mean a homomorphism from the dynamicalsystem (X, φ) to (Y, ψ) (i.e., a continuous map with α φ = ψ α) such that φ(y) = z.

Let (X, φ, y) be an essentially minimal t.d. dynamical system and C(X)⋊φ Z denotethe associated crossed product C*-algebra as described in [8]. Set C(X) ⋊φ Z =A(X, φ, y). In fact, the homeomorphism φ of X induces the automorphism C(φ−1) ofC(X) defined by C(φ−1)( f ) = f φ−1, f ∈ C(X). Then Z acts on C(X) by means of thisautomorphism and one considers the resulting crossed product C*-algebra C(X) ⋊φ Z.See [12] for the definition of (discrete) crossed products of C*-algebras. Also, eachmorphism in DS induces a morphism in the category of unital C*-algebras with unital∗-homomorphisms, C∗1.

Proposition 3.2. The mapA : DS→ C∗1, as defined above, is a contravariant functor.

Definition 3.3. Let X be a totally disconnected metrizable compact space, and letX = (X, φ, y) be an essentially minimal t.d. dynamical system. Denote by AF (X) theC*-subalgebra ofA(X) = C(X) ⋊φ Z generated by C(X) and u · C0(X \ y) where u isthe canonical unitary determining φ in C(X) ⋊φ Z. By [9, Theorem 3.3],AF (X) is anAF algebra.

Theorem 3.4. The mapAF : DS→ AF, as defined above, is a contravariant functor.

Remark 3.5. The functor AF : DS → AF is faithful. This is because for eachmorphism α : X1 → X2 as in the proof of Proposition 3.4, we have AF (α)(C(Y)) =A(α)(C(Y)) ⊆ C(X) and AF (α)( f ) = f α, f ∈ C(Y). The functor AF is nota strong classification functor (by Theorem 3.6 below). Therefore, AF is not a fullfunctor, since each full and faithful functor is a a strong classification functor (by [1,Lemma 5.10]). Similarly, the functor A : DS → C∗1 is faithful but not full. Also, it isnot a strong classification functor.

Recall that a minimal dynamical system (X, φ) is called a Cantor minimal system ifX is a compact metrizable space with a countable basis of clopen subsets and X has noisolated points (i.e., X is homeomorphic to the Cantor set). See [7] for the definitionof the notion of strong orbit equivalence.

Theorem 3.6. Let (X, φ) and (Y, ψ) be Cantor minimal systems. Let y and z be arbitrarypoints in X and Y, respectively. Then the following statements are equivalent:(1) (X, φ) and (Y, ψ) are strong orbit equivalent;(2) K0(X, φ) is order isomorphic to K0(Y, ψ) by a map preserving the distinguished

order unit;

19


(3) C(X) ⋊φ Z C(Y) ⋊ψ Z;(4) AF (X, φ, y) AF (Y, ψ, z) in AF.

Proof. The theorem follows from [7, Theorem 2.1] and [8, Theorem 5.3].

Giving a functorial formulation of the preceding theorem by defining a suitablenotion of morphism between Cantor minimal systems such that isomorphism coincideswith strong orbit equivalence is still an open question.

[1] M. Amini, G. A. Elliott, and N. Golestani, The category of Bratteli diagrams, Canad. J. Math. 67(2015), 990–1023.

[2] M. Amini, G. A. Elliott, and N. Golestani, The category of ordered Bratteli diagrams,arXiv:1509.07246, 2015, 31 pages.

[3] M. Amini, G. A. Elliott, and N. Golestani, The category of Cantor minimal systems, underpreparation, 12 pages.

[4] O. Bratteli, Inductive limits of finite dimensional C∗-algebras, Trans. Amer. Math. Soc. 171(1972), 195–234.

[5] G. A. Elliott, On the classification of inductive limits of sequences of semisimple finite dimensionalalgebras, J. Algebra 38 (1976), 29–44.

[6] G. A. Elliott, Towards a theory of classification, Adv. Math. 223 (2010), 30–48.[7] R. Giordano, I. F. Putnam, and C. F. Skau, Topological orbit equivalence and C∗-crossed products,

J. Reine Angew. Math. 469 (1995), 51–111.[8] R. H. Herman, I. F. Putnam, and C. F. Skau, Ordered Bratteli diagrams, dimension groups, and

topological dynamics, Internat. J. Math. 3 (1992), no. 6, 827–864.[9] I. F. Putnam, C*-algebras associated with minimal homeomorphisms of the Cantor set, Pacific J.

Math. 136 (1989), 329–353.[10] A. V. Versik, Uniform algebraic approximations of shift and multiplication operators, Dokl. Akad.

Nauk SSSR 259 (1981), no.3, 526–529. English translation: Sov. Math. Dokl. 24 (1981), 97–100.[11] N. E. Wegge-Olsen, K-Theory and C∗-Algebras, The Clarendon Press, New York, 1993.[12] D. P. Williams, Crossed Products of C*-Algebras, Vol. 134, American Mathematical Society,

2007.

M. Amini,Tarbiat Modares University and IPMe-mail: [email protected]

G. A. Elliott,University of Torontoe-mail: [email protected]

N. Golestani,Tarbiat Modares University and IPMe-mail: [email protected]

20



January 20–21, 2016

Stability of approximate dual fusion frames in Hilbert spaces

F. Arabyani Neyshaburi∗ and A. Arefijamaal

AbstractIn this paper we consider the notion of approximate duals for fusion frames. Then we study the stabilityof dual and approximate dual fusion frames under perturbations. Also we give a sufficient condition for afusion frame to be an approximate dual of its dual.

2010 Mathematics subject classification: Primary 42C15, Secondary 41A58.Keywords and phrases: fusion frames; approximate duals.

1. Introduction

Throughout this paperH denotes a separable Hilbert space, I is a countable index setand πV denotes the orthogonal projection onto closed subspace V . Recall that if Wii∈Iis a family of closed subspaces of a separable Hilbert space H and ωii∈I a family ofweights, i.e. ωi > 0, i ∈ I, then (Wi, ωi)i∈I is called a fusion frame forH if there existthe constants 0 < A ≤ B < ∞ such that

A∥ f ∥2 ≤∑i∈I

ω2i ∥πWi f ∥2 ≤ B∥ f ∥2, ( f ∈ H). (1)

The constants A and B are called the fusion frame bounds. If we only have the upperbound in (1) we call (Wi, ωi)i∈I a Bessel fusion sequence. A fusion frame is calledA-tight, if A = B, and Parseval if A = B = 1. For a Bessel fusion sequence (Wi, ωi)i∈IofH , the synthesis operator TW :

∑i∈I ⊕Wi → H is defined by

TW( fii∈I) =∑i∈I

ωi fi, ( fii∈I ∈∑i∈I

⊕Wi),

and its adjoint operator T ∗W : H → ∑i∈I ⊕Wi, is called the analysis operator. For every

fusion frame W = (Wi, ωi)i∈I , the fusion frame operator S W : H → H is defined byS W f =

∑i∈I ω

2i πWi f is a bounded, invertible as well as positive operator. In general, a

Bessel fusion sequence (Vi, νi)i∈I is called dual of (Wi, ωi)i∈I if

f =∑i∈I

ωiνiπVi S−1W πWi f , ( f ∈ H). (2)

∗ speaker

21

F. Arabyani Neyshaburi and A. Arefijamaal

It is clear that (S −1W Wi, ωi)i∈I , is a dual fusion frame of W, is called the canonical

dual of W. In [10], it is proved that a Bessel fusion sequence (Vi, υi)i∈I is adual of (Wi, ωi)i∈I if and only if TVϕvwT ∗W = IH , where the bounded operatorϕvw :

∑i∈I

⊕Wi →

∑i∈I

⊕Vi is given by

ϕvw( fii∈I) = πVi S−1W fii∈I . (3)

However dual fusion frames play a key role in fusion frame theory, their explicit com-putations seem rather intricate. In this respect, we consider the notion of approximatedual for fusion frames and discuss on the stability of approximate dual fusion frames.

2. Stability of approximate duals

The stability of frames under perturbations, which is important in practice, has beenstudied by many authors, for example the stability of approximate dual of frames andg-frames can be found in [5, 7]. In this section, we survey the stability of approximatedual fusion frames. First we recall that Λ = λi ∈ B(H ,Hi) : i ∈ I is a g-frame forHwith respect to Hi : i ∈ I if there exist two positive numbers A and B such that

A∥ f ∥2 ≤∑i∈I

∥λi f ∥2 ≤ B∥ f ∥2, ( f ∈ H).

Moreover Λ is called g-Bessel sequence if in (4)only the upper inequality is hold [11].If Λ and Γ are two g-Bessel sequences, the operator S ΓΛ : H → H is defined as

S ΓΛ f =∑i∈I

Γ∗i λi f .

S ΓΛ is a well defined, bounded and S ∗ΓΛ= SΛΓ, see [8]. Approximate duality

introduced in [5] for discrete frames. Also in [7] approximate duality of g-frameswas defined as follows:

Definition 2.1. Two g-Bessel sequence Λ and Γ are approximate dual g-frames if∥IH − S ΓΛ < 1∥ or ∥IH − SΛΓ < 1∥,

We define the notion of approximate dual fusion frames similarly.

Definition 2.2. Let (Wi, ωi)i∈I be a Bessel fusion sequence. A Bessel fusion sequence(Vi, υi)i∈I is called an approximate dual of (Wi, ωi)i∈I if

∥IH − TVϕvwT ∗W∥ < 1.

Putting

ψvw = TVϕvwT ∗W . (4)

Then, we have the following reconstruction formula

f =∑i∈I

(ψvw)−1ωiυiπVi S−1W πWi f =

∞∑n=0

(I − ψvw)nψvw f , ( f ∈ H).

22

Approximate duals

Remark 2.3. Let λi = υiπVi and Γi = ωiS −1W πWi . Then Λ = λii∈I and Γ = Γii∈I

are g-Bessel esquences and SΛΓ = TVϕvwT ∗W , so ∥IH − SΛΓ∥ = ∥TVϕvwT ∗W∥. Thismeans that (Vi, υi)i∈I is an approximate dual of (Wi, ωi)i∈I if and only if Λ is anapproximate g-dual of Γ. Hence approximate duality fusion frames is a special caseof approximate duality g-frames introduced in [7]. Also approximate duality of fusionframes in the other notion has been considerd in Corollary 2.4 and Proposition 2.14 in[7] and some results about perturbations of approximate dual fusion frames has beenobtained in Corollary 3.3 and Corollary 3.9 in [7].

The following proposition describes approximate duality of fusion frames withrespect to local frames, see [1].

Proposition 2.4. Let e j j∈J be an orthonormal basis of H . The Bessel sequenceV = (Vi, υi)i∈I is an approximate dual of W = (Wi, ωi)i∈I if and only if υiπVi e ji∈I, j∈J

is an approximate dual of ωiπWi S−1W e ji∈I, j∈J .

Definition 2.5. Let Wii∈I and Wii∈I be closed subspaces in H . Also let ωii∈I bea sequence of positive numbers and ϵ > 0. We call (Wi, ωi)i∈I a ϵ-perturbation of(Wi, ωi)i∈I whenever for every f ∈ H ,∑

i∈I

ω2i ∥(πWi

− πWi ) f ∥2 < ϵ.

Theorem 1. Let V = (Vi, υi)i∈I be an approximate dual of a Bessel fusion sequenceW = (Wi, ωi)i∈I . Also let (Ui, υi)i∈I be a ϵ-perturbation of V , such that

ϵ <

1 − ∥IH − ψvw∥√B∥S −1

W ∥

2

, (5)

where B and D are the upper bounds of W and V , respectively. Then (Ui, υi)i∈I isalso an approximate dual of W. In particular, if W is a Parseval fusion frame andchoose V = W, then the result holds for ϵ < 1.

Example 2.6. Consider

W1 = R2 × 0, W2 = 0 × R2, W3 = span(1, 0, 0),

V1 = span(0, 1, 0), V2 = 0 × R2, V3 = span(1, 0, 0).

Then ∥S −1W ∥ = 1 and ∥IH − ψvw∥ = 1

2 . Hence, the Bessel sequence V = Vi3i=1 is anapproximate dual of fusion frame W = Wi3i=1. Now, if we take

U1 = V1, U2 = V2, U3 = span(α, β, 0),

where 12 ≤ α < 1 and 0 ≤ β ≤ 1

100 , then U = Uii∈I is a ϵ-perturbation of V withϵ < 1

8 . Hence, by Theorem 1, U is also an approximate dual of W.

The next result is obtained immediately from Theorem 1.

23

F. Arabyani Neyshaburi and A. Arefijamaal

Corollary 2.7. Let (Vi, υi)i∈I be an alternate dual of a Bessel fusion sequence W =(Wi, ωi)i∈I . Also let (Ui, υi))i∈I be a ϵ-perturbation of V, and

√ϵB ≤ 1

∥S −1W ∥

, (6)

where B is the upper bound of W. Then (Ui, υi)i∈I is an approximate dual of W.

Theorem 2.8. Let V = (Vi, υi)i∈I be an approximate dual of a Bessel fusion sequenceW = (Wi, ωi)i∈I . Also if Uii∈I is a ϵ-perturbation of W with sufficiently small boundϵ > 0, then (Vi, υi)i∈I is also an approximate dual of U = (Ui, ωi)i∈I .

Example 2.9. Consider

V1 = R3, V2 = 0 × R2, V3 = span(1, 0, 0).

Then V = Vi3i=1 is a dual of Parseval fusion frame W = Wi3i=1, in which

W1 = span(0, 0, 1), W2 = span(0, 1, 0), W3 = span(1, 0, 0).On the other hand, letting

U1 = W1, U2 = W2, U3 = span(1, 150, 0).

Then Uii∈I is a ϵ-perturbation of W with ϵ < 0.02. Using the fact that

0.02 <1 −√

2∥IH − S −1U ∥√

2∥S −1U ∥

.

we obtain V is an approximate dual of Uii∈I by Theorem 2.8.

Many concepts of the classical frame theory have not been generalized to fusionframes. For example, in the duality discussion, if V = (Vi, υi)i∈I is a dual of fusionframe W = (Wi, ωi)i∈I , then W is not a dual of V , see Example 2.2 in [10]. The nexttheorem gives a sufficient condition for a fusion frame is approximate dual of its dual.

Theorem 2.10. Let (Vi, υi)i∈I be a dual of fusion frame (Wi, ωi)i∈I such that

∥S −1W − S −1

V ∥ < ∥S W∥−1/2∥S V∥−1/2.

Then (Wi, ωi)i∈I is an approximate dual of (Vi, υi)i∈I .

The fusion frame W in Example 2.9 is not a dual of V , however, a straightforwardcalculation shows that

∥S −1V − S −1

W ∥ =12, ∥S V∥ = 2.

Hence, W is an approximate dual of V by Theorem 2.10. It is worth noticing that,unlike discrete frames, ψ−1

wvWi3i=1 is not dual of Vi3i=1. Indeed ψ−1wv = 2IH and so∑

i∈I

πψ−1wvWi

S −1V πVi =

12

IH .

24

Approximate duals

However, it can be considered as a Q-dual of Vi3i=1 with

Q( fii∈I) = ψ−1vwπVi S

−1W fii∈I , ( fi ∈

∑i∈I

⊕Wi).

More precisely, let e j j∈J be an orthonormal basis for H . By Proposition 2.4, wecan derive that the sequence G = υiπVi e ji∈I, j∈J is an approximate dual of F =ωiπWi S

−1W e ji∈I, j∈J . Hence (TGT ∗F)−1G is a dual of ωiπWi S

−1W e ji∈I, j∈J . On the other

hand, TGT ∗F = TVϕvwT ∗W = ψvw. Hence (ψ−1vwVi, υi)i∈I is a Q-dual of (Wi, ωi)i∈I by

Theorem 3.12 in [9].

[1] A. A. Arefijamaal, F. Arabyani Neyshaburi, Characterization of dual Riesz fusion bases byusing approximate duals. Arxiv.

[2] M. S. Asgari, On the Riesz fusion bases in Hilbert spaces, Egypt. Math. Soc. 21 (2013), 79-86.[3] P. G. Casazza, G. Kutyniok, Frames of subspaces, Contemp. Math. 345 (2004), 87-114.[4] P. G. Casazza, G. Kutyniok and S. Li, Fusion frames and distributed processing, Appl. Comput.

Harmon. Anal. 25 (1) (2008), 114-132.[5] O. Christensen, R. S. Laugesen, Approximately dual frames in Hilbert spaces and applica-

tions to Gabor frames, Sampl. Theory Signal Image Process. 9 (3) (2010), 77-89.[6] O. Christensen, Frames and Bases: An Introductory Course, Birkhäuser, Boston, 2008.[7] A. Khosravi, M. Mirzaee Azandaryani, Approximate duality of g-frames in Hilbert spaces,

Acta Math. Sci. Ser. B Engl. Ed. 34 (3) (2014), 639-652.[8] A. Khosravi, K. Musazadeh, Fusion frames and g-frams, Math. Anal. Appl. 342 (2008), 1068-

1083.[9] S. B. Heineken, P. M. Morillas, A. M. Benavente, M. I. Zakowicz, Dual fusion frames, Arch.

Math. 103 (2014), 355-365.[10] E. Osgooei, A. A. Arefijamaal, Compare and contrast between duals of fusion and discrete

frames, Submitted.[11] W. Sun, G-frames and g-Riesz bases, Math. Anal. Appl. 322 (2006), 437-452.

F. Arabyani Neyshaburi,Department of Mathematics and Computer Sciences,University of Hakim Sabzevari,City Sabzevar, Iran.e-mail: [email protected]

A. Arefijamaal,Department of Mathematics and Computer Sciences,University of Hakim Sabzevari,City Sabzevar, Iran.e-mail: [email protected]

25



January 20–21, 2016

On stability of generalized dual Banach frames

M. S. Asgari∗ and F. Enayati

Abstract

In this paper, we study the stability of dual, canonical dual, pseudo-dual and approximate dual for Besselsequences with respect to a BK-space. Some new conditions for the stability of generalized dual Banachframes have been given.

2010 Mathematics subject classification: Primary 42C99, Secondary 46B99, 46C99..Keywords and phrases: Xd-Frames, Banach frames, approximate duals, pseudo-duals..

1. Introduction

Throughout this paper X will denote a separable Banach space over the scalar field F(R or C), X∗ the dual space of X, I the countable index set that has been well-ordered,In, n ∈ N the subset of the first n indices in I.

A BK-space Xd on I is a Banach space of sequences c = cii∈I ∈ CI with theproperty that the coordinate linear functionals c → ci, i ∈ I are continuous on Xd. Weshall require that the canonical unit vectors ei, i ∈ I form a Schauder basis for Xd. Bya result in [3], the dual space X∗d of Xd is also a BK-space of sequences d = dii∈I ⊆ Csuch that d(c) =

∑i∈I cidi for all c ∈ Xd and d ∈ X∗d. We also impose several more

crucial assumptions on Xd. Specifically, we require that Xd be reflexive, the canonicalunit vectors ei, i ∈ I form a Schauder basis for X∗d as well, if d = dii∈I ∈ CI and∑

i∈I cidi converges for every c ∈ Xd then d ∈ X∗d, and if the series converges for alld ∈ X∗d then c ∈ Xd.

Definition 1.1. Suppose that Xd is a BK-space on I. Let xii∈I ⊆ X and fii∈I ⊆ X∗.Then

(i) fii∈I is called an Xd-frame for X with bounds A and B if(a) fi(x)i∈I ∈ Xd for each x ∈ X, and(b) there exist positive constants A and B with 0 < A ≤ B < ∞ such that

A∥x∥X ≤ ∥ fi(x)i∈I∥Xd ≤ B∥x∥X , ∀x ∈ X, (1)

∗ speaker

26

M. S. Asgari and F. Enayati

If at least (a) and the upper condition in (b) are satisfied, fii∈I is called an Xd-Bessel sequence for X. If there exists a bounded linear operator S : Xd → X suchthat S ( fi(x)i∈I) = x for each x ∈ X, then ( fii∈I , S ) is called a Banach frame forX with respect to Xd. The operator S is said to be the reconstruction operator.Conditions (a) and (b) allow the definition of the analysis operator U : X → Xd,Ux = fi(x)i∈I . The reconstruction and analysis operators determine the Banachframe for X in the following sense: if ( fii∈I , S ) is a Banach frame for X withrespect to Xd then, S U = IdX and fi = U∗ei, where eii∈I ⊂ X∗d is the Schauderbasis for X∗d and U∗ : X∗d → X∗ is the transpose of U.

(ii) xii∈I is called an X∗d-frame for X∗ with bounds A and B if(a) f (xi)i∈I ∈ X∗d for each f ∈ X∗, and(b) there exist positive constants A and B with 0 < A ≤ B < ∞ such that

A∥ f ∥X∗ ≤ ∥ f (xi)i∈I∥X∗d ≤ B∥ f ∥X∗ , ∀ f ∈ X∗, (2)

If at least (a) and the upper condition in (b) are satisfied, xii∈I is called an X∗d-Bessel sequence for X∗. If there exists a bounded linear operator T : X → Xd

such that T ∗( f (xi)i∈I) = f for each f ∈ X∗, then (xii∈I ,T ) is called a Banachframe for X∗ with respect to X∗d. The operator T is said to be the reconstructionoperator. Conditions (a) and (b) allow the definition of the synthesis operatorV : Xd → X, Vc =

∑i∈I cixi. The reconstruction and synthesis operators

determine the Banach frame for X∗ in the following sense: if (xii∈I ,T ) is aBanach frame for X∗ with respect to X∗d then, VT = IdX and xi = Vei, whereeii∈I ⊂ Xd is the Schauder basis for Xd (see [4]).

Definition 1.2. Suppose that fii∈I , xii∈I are Xd-Bessel and X∗d-Bessel sequences forX, X∗ with analysis and synthesis operators U,V respectively. Then(i) (xii∈I , fii∈I) is called dual Banach frame for X with respect to Xd, if VU = IdX .(ii) (xii∈I , fii∈I) is called a pseudo-dual Banach frame for X with respect to Xd, if

VU is a bijection on X.(iii) (xii∈I , fii∈I) is called an approximate dual Banach frame for X with respect to

Xd, if ∥IdX − VU∥ < 1.

2. Main Results

In this section we generalize some results of Christensen [1] to the situation ofdual Banach frames. We also show that generalized dual Banach frames are stableunder small perturbations of the Banach frame elements so that the perturbation resultsobtained in [2] is a special case of it.

Proposition 2.1. Let fii∈I , xii∈I be Xd-Bessel and X∗d-Bessel sequences for X, X∗ withanalysis and synthesis operators U,V respectively. Then the following statements areequivalent:(i) (xii∈I , fii∈I) is a dual Banach frame for X with respect to Xd.(ii) f =

∑i∈I f (xi) fi, ∀ f ∈ X∗.

27


(iii) f (x) =∑

i∈I fi(x) f (xi), ∀x ∈ X, f ∈ X∗.

Proof. The equivalence of (i), (ii) and (iii) follows immediately from the definition.

Proposition 2.2. Let fii∈I , xii∈I be Xd-Bessel and X∗d-Bessel sequences for X, X∗ withanalysis and synthesis operators U,V respectively. Then(i) If ( fii∈I , S ) is a Banach frame for X with respect to Xd, then (S eii∈I , fii∈I) is

a dual Banach frame for X with respect to Xd, where eii∈I ⊂ Xd is the Schauderbasis for Xd.

(ii) If (xii∈I ,T ) is a Banach frame for X∗ with respect to X∗d, then (xii∈I , T ∗eii∈I) isa dual Banach frame for X with respect to Xd, where eii∈I ⊂ X∗d is the Schauderbasis for X∗d.

Proof. This follows immediately from the Proposition 2.1. The dual Banach frames in Proposition 2.2 are said to be the canonical dual Banach

frames for X with respect to Xd.

Proposition 2.3. Let (xii∈I , fii∈I) be a dual Banach frame for X with respect to Xd.Then ( fii∈I ,V) and (xii∈I ,U) are Banach frames for X, X∗ with respect to Xd, X∗drespectively, where U,V are the analysis and synthesis operators of fii∈I , xii∈I .

Proof. The assumption assures that

∥V∥−1∥x∥X = ∥V∥−1∥VU(x)∥X ≤ ∥U(x)∥Xd = ∥ f (xi)i∈I∥Xd ≤ ∥U∥ ∥x∥X ,

for all x ∈ X. Now VU = IdX implies that ( fii∈I ,V) is a Banach frame for X withrespect to Xd. Similarly, we can show that (xii∈I ,U) is also a Banach frame for X∗

with respect to X∗d.

Remark 2.4. Note that if (xii∈I , fii∈I) is a pseudo-dual Banach frame for X withrespect to Xd with analysis and synthesis operators U,V respectively. Then

x =∑i∈I

fi(x)(VU)−1xi ∀ x ∈ X.

Thus ((VU)−1xii∈I , fii∈I) is a dual Banach frame for X with respect to X∗d. Now theProposition 2.3 shows that

( fii∈I , (VU)−1V)

is a Banach frame for X with respectto Xd. By symmetry

(xii∈I ,U(VU)−1) is also a Banach frame for X∗ with respect toX∗d. Furthermore, if (xii∈I , fii∈I) is an approximate dual Banach frame for X withrespect to Xd. Then from the condition ∥IX − VU∥ < 1 implies that the operator VU isa bijection on X. Thus every approximate dual Banach frame is a pseudo-dual Banachframe.

The next theorem shows that approximate dual Banach frames are stable undersmall perturbations of the Banach frame elements so that Theorem 2.2 obtained in [2]is a special case of it.

Theorem 2.5. Let ( fii∈I , S ) be a Banach frame for X with respect to Xd with theanalysis operator U. Assume that gii∈I ⊆ X∗ and there exist λ, µ ≥ 0 such that

28


(i) 2(λ∥U∥ + µ)∥S ∥ < 1.(ii) ∥ fi(x) − gi(x)i∈I∥Xd ≤ λ∥ fi(x)i∈I∥Xd + µ∥x∥X ,for all x ∈ X. Then there exists a reconstruction operator S 1 such that (gii∈I , S 1) isa Banach frame for X with respect to Xd and (S eii∈I , gii∈I) and (S 1eii∈I , fii∈I)are approximate dual Banach frames for X with respect to Xd respectively, whereeii∈I ⊂ Xd is the Schauder basis for Xd.

Proof. Let U1 be the analysis operator of gii∈I , from the hypotheses we have

∥U1x∥Xd ≤ ∥U1x − Ux∥Xd + ∥Ux∥Xd ≤((λ + 1)∥U∥ + µ)∥x∥X ,

for all x ∈ X. This establishes the upper frame bound for gii∈I . On the otherhand, from S U = IX we have ∥IX − S U1∥ ≤ ∥S ∥ ∥U − U1∥ < 1 which follows that(S eii∈I , gii∈I) is an approximate dual Banach frame for X with respect to Xd and so∥(S U1)−1∥ ≤ 1

1−(λ∥U∥+µ)∥S ∥ . If we set S 1 = (S U1)−1S , then S 1U1 = IX , which implies

that (gii∈I , S 1) is a Banach frame for X with respect to Xd and ∥S 1∥ ≤ ∥S ∥1−(λ∥U∥+µ)∥S ∥ .

Finally, we obtain

∥IX − S 1U∥ = ∥S 1U1 − S 1U∥ ≤ ∥S 1∥∥U1 − U∥ ≤ (λ∥U∥ + µ)∥S ∥1 − (λ∥U∥ + µ)∥S ∥ < 1.

This completes the proof.

Corollary 2.6. Let (xii∈I , fii∈I) be a dual Banach frame for X with respect to Xd withthe analysis and synthesis operators U,V. Assume that gii∈I is a sequence in X∗ andthere exist λ, µ ≥ 0 such that(i) 2(λ∥U∥ + µ)∥V∥ < 1.(ii) ∥ fi(x) − gi(x)i∈I∥Xd ≤ λ∥ fi(x)i∈I∥Xd + µ∥x∥X .Then (xii∈I , gii∈I) is an approximate dual Banach frame for X with respect to Xd.

Proof. The proof is similar to Theorem 2.5.

Theorem 2.7. Let (xii∈I , fii∈I) be a dual Banach frame for X with respect to Xd withthe analysis and synthesis operators U,V. Assume that yii∈I is a sequence in X andthere exist λ, µ ≥ 0 such that(i) (1 + ∥V∥∥U∥)(λ + µ∥U∥) < 1.(ii)

∥∥∥∑i∈I ci(xi − yi)

∥∥∥X ≤ λ

∥∥∥∑i∈I cixi

∥∥∥X + µ∥c∥Xd ,

for all c ∈ Xd. Then (yii∈I , fii∈I) is an approximate dual Banach frame for X withrespect to Xd. Furthermore, there exists a Xd-Bessel sequence gii∈I for X such that(xii∈I , gii∈I) is an approximate dual Banach frame for X with respect to Xd.

Proof. The hypotheses given imply that the series∑

i∈I fi(x)yi is convergent in X forevery x ∈ X. Thus the operator Λ : X → X defined by Λx =

∑i∈I fi(x)yi is bounded

and holds

∥x − Λx∥X ≤ λ∥x∥X + µ∥Ux∥Xd ≤ (λ + µ∥U∥)∥x∥X .From this we have ∥IX − Λ∥ < 1, which follows that (yii∈I , fii∈I) is an approximatedual Banach frame for X with respect to Xd and ∥Λ−1∥ ≤ 1

1−(λ+µ∥U∥) . Therefore if we

29


define gi = (Λ−1)∗( fi), i ∈ I, then (yii∈I , gii∈I) is a dual Banach frame for X withrespect to Xd. Let U1 be the analysis operator of gii∈I . Then we have

∥IX − VU1∥ ≤ ∥V∥∥U − U1∥ ≤ ∥V∥∥U∥∥IX − Λ−1∥

≤ ∥V∥∥U∥∥Λ−1∥∥IX − Λ∥ ≤ ∥V∥∥U∥λ + µ∥U∥

1 − (λ + µ∥U∥) < 1.

Therefore (xii∈I , gii∈I) is an approximate dual Banach frame for X with respect toXd.

Acknowledgement

The authors are grateful to the reviewers for their accurate reading and their helpfulsuggestions.

[1] O. Christensen, An introductory course of frames and bases, Birkhauser, Boston, 2008.[2] O. Christensen, C. Heil, Perturbations of Banach frames and atomic decompositions, Math.

Nachr. 185 (1997) 33-47.[3] L. V. Kantorovich, G. P. Akilov, Functional Analysis in Normed Spaces, Macmillan, New York,

1964.[4] S. S. Karimizad, M. S. Asgari, On the duality of Banach frames, U. P. B. Sci. Bull., Series A. 77

(3) (2015) 167-182.

M. S. Asgari,Department of Mathematics, Faculty of Science,Central Tehran Branch, Islamic Azad University,Tehran, Iran.e-mail: [email protected]; [email protected]

F. Enayati,Department of Mathematics, Faculty of Science,Central Tehran Branch, Islamic Azad University,Tehran, Iran.e-mail: [email protected]

30



January 20–21, 2016

Generalized Riesz-dual sequences in Hilbert spaces

M. S. Asgari and F. Enayati∗

Abstract

The R-dual sequences of a frame introduced by Casazza, Kutyniok and Lammers in [3]. In this articlewe introduce the R-dual sequences for a g-frame. We show that in fact for each g-Bessel sequence wecan construct a corresponding generalized R-dual sequence with a kind of duality relation between them.This construction is used to prove duality principles in g-frame theory, which can be regarded as generalversions of several well-known duality principles for g-frames.

2010 Mathematics subject classification: Primary 41A58, Secondary 42C15, 42C05, 46C99..Keywords and phrases: G-orthonormal bases; g-frames; generalized Riesz-dual sequences; Riesz-dualsequence..

1. Introduction

Generalized Riesz-dual sequences or simply g-R-dual sequences are natural general-izations of R-dual sequences which provide a powerful tool in the analysis of dual-ity relations in general g-frame theory. The purpose of this paper is to analyze theconcept of Riesz-dual sequence for g-frames. We give characterizations of g-R-dualsequences and prove that g-R-dual sequences share many useful properties with R-dualsequences.

Throughout this paperH andK are two separable Hilbert spaces, Vii∈I is a familyof closed subspaces of K and B(H ,Vi) denote the collection of all bounded linearoperators fromH into Vi for all i ∈ I.

Recall [7], that a family Λ = Λi ∈ B(H ,Vi) : i ∈ I is a generalized frame orsimply a g-frame forH with respect to Vii∈I if there exist constants 0 < C ≤ D < ∞such that:

C∥ f ∥2 ≤∑i∈I

∥Λi f ∥2 ≤ D∥ f ∥2, ∀ f ∈ H . (1)

The constants C and D are called g-frame bounds. If only the right-hand inequality of(1) is required, we call it a g-Bessel sequence.

Now we briefly recall the definitions and basic properties of g-bases, g-orthonormalbases and g-Riesz bases. For more information we refer to [1, 2, 4, 5].∗ speaker

31


Definition 1.1. Let Ξi ∈ B(H ,Wi)| i ∈ I be a sequence of operators. Then(i) Ξii∈I is a generalized Schauder basis or simply a g-basis forH with respect to

Wii∈I , if for all f ∈ H there exist unique vectors gi ∈ Wi, i ∈ I with

f =∑i∈I

Γ∗i gi. (2)

If Ξii∈I is a g-basis only for its closed linear span, we call it a g-basic sequencewith respect to Wii∈I .

(ii) Ξii∈I is a g-complete set forH with respect to Wii∈I , ifH = spanΞ∗i (Wi)i∈I .(iii) Ξii∈I is a g-orthonormal system forH with respect to Wii∈I , if ΞiΞ

∗jg j = δi jg j

for all i, j ∈ I and g j ∈ W j.(iv) A g-complete and g-orthonormal system Ξii∈I is called a g-orthonormal basis

forH with respect to Wii∈I .

Definition 1.2. A sequence Γ = Γ j ∈ B(H ,W j)| j ∈ I is called a g-Riesz basis forHwith respect to W j j∈I , if Γ j j∈I is a g-complete set forH with respect to W j j∈I andthere exist constants 0 < A ≤ B < ∞ such that:

A∑j∈I

∥g j∥2 ≤∥∥∥∑

j∈I

Γ∗jg j

∥∥∥2 ≤ B∑j∈I

∥g j∥2, (3)

for all sequences g j j∈I ∈(∑

j∈I ⊕W j)ℓ2 . We define the g-Riesz basis bounds for Γ j j∈I

to be the largest number A and the smallest number B such that this inequality (3)holds. If Γ j j∈I is a g-Riesz basis only for spanΓ∗j(W j) j∈I , we call it a g-Riesz basicsequence forH with respect to W j j∈I .

Definition 1.3. Let Ξ = Ξii∈I and Ξ′ = Ξ′ii∈I be g-orthonormal bases for H withrespect to Wii∈I and Vii∈I respectively. The transition matrix from Ξ to Ξ′ is thematrix B = [Bi j] whose (i, j)-entry is Bi j = Ξ

′iΞ∗j for all i, j ∈ I. Then we have

B[ f ]Ξ = [ f ]Ξ′ where, [ f ]Ξ is the coordinate representation of an arbitrary vectorf ∈ H in the basis Ξ and similarly for Ξ′. The transition matrix from Ξ′ to Ξ isB−1 = B∗. Thus, if B∗ = [B∗i j] then B∗i j = (B ji)∗ = ΞiΞ

′∗j for all i, j ∈ I.

2. Main Results

In this section we define the g-R-dual sequence from a sequence of operators. Thenwe exactly characterize all sequences with lower g-frame bounds. Next, we obtain theg-frame conditions for a sequence of operators and its g-R-dual sequence.

Definition 2.1. Let Ξ = Ξii∈I and Ψ = Ψii∈I be g-orthonormal bases for H withrespect to Wii∈I and Vii∈I respectively. Let Λ = Λi : H → Vi| i ∈ I be such thatthe series

∑i∈I Λ

∗i g′i is convergent for all g′ii∈I ∈

(∑i∈I ⊕Vi

)ℓ2 . For all j ∈ I, let

ΓΛj : H → W j, ΓΛj =∑i∈I

Ξ jΛ∗iΨi. (4)

Then ΓΛj j∈I is called the generalized Riesz-dual sequence (g-R-dual sequence) for thesequence Λ with respect to (Ξ,Ψ).

32


Now, we give an algorithm to invert the process and calculate Λii∈I from thesequence ΓΛj j∈I .

Theorem 2.2. Let Ξ = Ξii∈I and Ψ = Ψii∈I be g-orthonormal bases for H withrespect to Wii∈I and Vii∈I respectively. Let Λii∈I be a g-Bessel sequence for Hwith respect to Vii∈I . Then, for all i ∈ I,

Λi =∑j∈I

Ψi(ΓΛj )∗Ξ j. (5)

In particular, this shows that Λii∈I is the g-R-dual sequence for ΓΛj j∈I with respectto (Ψ,Ξ).

Proof. The definition of ΓΛj j∈I implies that for every i, j ∈ I and g j ∈ W j

Ψi(ΓΛj )∗g j = Ψi(∑

k∈I

Ψ∗kΛkΞ∗jg j

)=

∑k∈I

ΨiΨ∗kΛkΞ

∗jg j

=∑k∈I

δikΛkΞ∗jg j = ΛiΞ

∗jg j.

Therefore Ψi(ΓΛj )∗ = ΛiΞ∗j . Now, for each f ∈ H we have

Λi f = Λi(∑

j∈I

Ξ∗jΞ j f)=

∑j∈I

ΛiΞ∗jΞ j f =

∑j∈I

Ψi(ΓΛj )∗Ξ j f .

The next result gives a characterization of g-frame sequences which keeps the

information about the g-frame bounds.

Proposition 2.3. Let Λ = Λi ∈ B(H ,Vi) : i ∈ I. Then the following conditions areequivalent.(i) Λ = Λii∈I is a g-frame sequence with respect to Vii∈I with g-frame bounds A

and B.(ii) The synthesis operator T ∗

Λis well defined on

(∑i∈I ⊕Vi

)ℓ2 such that:

A∥g′∥2ℓ2 ≤ ∥T ∗Λg′∥2 ≤ B∥g′∥2

ℓ2 , ∀ g′ ∈ N⊥T ∗Λ.

Proof. The proof is similar to the proof of this theorem in frame situation.

Theorem 2.4. Let Λ = Λii∈I be a g-Bessel sequence forH with respect Vii∈I . Thenfor every g j j∈I ∈

(∑j∈I ⊕W j

)ℓ2 , g′ii∈I ∈

(∑i∈I ⊕Vi

)ℓ2 we have∥∥∥∥∑

j∈I

(ΓΛj )∗g j

∥∥∥∥2=

∑i∈I

∥Λi f ∥2 and∥∥∥∥∑

i∈I

Λ∗i g′i∥∥∥∥2=

∑j∈I

∥ΓΛj h∥2,

where f =∑

j∈I Ξ∗jg j and h =

∑i∈I Ψ

∗i g′i .

33


Proof. Using Definition 2.1, we have∥∥∥∥∑j∈I

(ΓΛj )∗g j

∥∥∥∥2=

∥∥∥∥∑j∈I

(∑i∈I

Ξ jΛ∗iΨi

)∗g j

∥∥∥∥2=

∥∥∥∥∑i∈I

Ψ∗iΛi f∥∥∥∥2

=⟨∑

i∈I

Ψ∗iΛi f ,∑j∈I

Ψ∗jΛ j f⟩=

∑i∈I

∑j∈I

⟨Λi f ,ΨiΨ∗jΛ j f ⟩

=∑i∈I

∑j∈I

⟨Λi f , δi jΛ j f ⟩ =∑i∈I

∥Λi f ∥2.

Similarly, the second claim follows from Theorem 2.2.

Corollary 2.5. Let Λ = Λii∈I be a g-Bessel sequence forH with respect Vii∈I . Thenfor any f ∈ H we have ∥T ∗

ΓΛ

([ f ]Ξ

)∥ = ∥TΛ f ∥ℓ2 and ∥T ∗Λ

([ f ]Ψ

)∥ = ∥TΓΛ f ∥ℓ2 .

Proof. This follows immediately from the Theorem 2.4.

Theorem 2.6. Let Λ = Λii∈I be a g-Bessel sequence forH with respect to Vii∈I withg-R-dual sequence ΓΛj j∈I with respect to (Ξ,Ψ). Then the following statements hold.(i) f ∈ (

span(ΓΛj )∗(W j) j∈I)⊥ if and only if [ f ]Ψ ∈ NT ∗

Λ.

(ii) f ∈ (spanΛ∗j(V j) j∈I

)⊥ if and only if [ f ]Ξ ∈ NT ∗ΓΛ

.

Proof. Let f ∈ H . First for each j ∈ J and g j ∈ W j we observe that

⟨ f , (ΓΛj )∗g j⟩ =∑i∈J

⟨ f ,Ψ∗iΛiΞ∗jg j⟩ =

⟨∑i∈J

Λ∗iΨi f ,Ξ∗jg j⟩=

⟨T ∗Λ([ f ]Ψ),Ξ∗jg j

⟩.

Since Ξ = Ξ j j∈J is a g-orthonormal basis forH with respect to W j j∈I ,⟨T ∗Λ([ f ]Ψ),Ξ∗jg j

⟩= 0,

for all j ∈ I and g j ∈ W j, if and only if T ∗Λ

([ f ]Ψ) = 0. Thus, by definition of T ∗Λ

,f ∈ (

span(ΓΛj )∗(W j) j∈I)⊥ if and only if [ f ]Ψ ∈ NT ∗

Λ. Similarly, the second claim

follows from Theorem 2.2. The next result shows a kind of equilibrium between a sequence of operators and

its R-dual sequence. It can be viewed as a general version of Proposition 13 in [3] and[6].

Corollary 2.7. The following conditions are equivalent.(i) Λ = Λii∈I is a g-frame sequence with respect to Vii∈I with g-frame bounds A

and B.(ii) ΓΛj j∈I is a g-frame sequence with respect to W j j∈I with g-frame bounds A and

B.

Proof. Let f ∈ H . By Corollary 2.5 we have ∥T ∗Λ

([ f ]Ψ

)∥ = ∥TΓΛ f ∥ℓ2 . Now, the claimfollows immediately from the Proposition 2.3 and Theorem 2.6.

The following result characterize those sequences ΓΛj j∈I , which are g-Riesz basicsequences, in term of properties of Λii∈I .

34


Theorem 2.8. The following conditions are equivalent.(i) Λ = Λii∈I is a g-frame forH with respect to Vii∈I with g-frame bounds A, B.(ii) ΓΛj j∈I is a g-Riesz basic sequence with respect to W j j∈I with g-frame bounds

A, B.

Proof. The equivalent of (i) and (ii) follows immediately from Theorem 2.4.

Acknowledgement

The authors are grateful to the reviewers for their accurate reading and their helpfulsuggestions.

[1] M. S. Asgari, Operator-valued bases on Hilbert spaces, J. Linear and Topological Algebra. 2 (4)(2013) 201-218.

[2] M. S. Asgari, H. Rahimi, Generalized frames for operators in Hilbert spaces, Infin. Dimens. Anal.Quantum. Probab. Relat. Top. 17 (2) (2014) 1450013 (20 pages).

[3] P. G. Casazza, G. Kutyniok and M. C. Lammers, Duality principles in frame theory, J. FourierAnal. Appl. 10 (2004) 383-408.

[4] X. Guo, G-bases in Hilbert spaces, Abstract and Applied Analysis, 2012 Article ID 923729, 14pages, doi:10.1155/2012/923729.

[5] X. Guo, Operator parameterizations of g-frames, Taiwanese J. Math. 18 (1) (2014) 313-328.[6] A. Ron, Z. Shen, WeylHeisenberg frames and Riesz bases in L2(Rd),Duke Math. J. 89 (1997)

237-282.[7] W. Sun, G-frames and g-Riesz bases, J. Math. Anal. Appl. 322 (2006) 437-452.

M. S. Asgari,Department of Mathematics, Faculty of Science,Central Tehran Branch, Islamic Azad University,Tehran, Iran.e-mail: [email protected] ; [email protected]

F. Enayati,Department of Mathematics, Faculty of Science,Central Tehran Branch, Islamic Azad University,Tehran, Iran.e-mail: [email protected]

35



January 20–21, 2016

Harmonic analysis meets Lie theory via shearlet group

V. Atayi∗ and R. Kamyabi-Gol

Abstract

We consider the shearlet group as a Lie group and compute its Lie algebra.

2010 Mathematics subject classification: Primary 42C40, Secondary 22E60.Keywords and phrases: shearlet group, Lie group, Lie algebra, exponential map, semidirect product.

1. Introduction

Lie algebras are favorite objects because they are a sort of "linearization" of Lie groupsso that reduce geometry to linear algebra. In the recent years, shearlet group which isthe locally compact group related to the shearlet transform has been investigated fromvarious points of view such as group theory, harmonic analysis and image processing,see [6],[5],[4],[3],[1]. However, so far, the shearlet group has not been considered asa Lie group. The authors show in [2] and [8] that shearlet group can be thought of a3-fold semidirect product (Lie) group, the fact that we use to extract its Lie algebra.

2. Preliminaries

Let G be a Lie group, p ∈ G and (ϕ,U) a chart of G with p ∈ U. Let γ : I → G be asmooth curve, where I ⊆ R is an interval containing 0 and γ(0) = p. We call two suchcurves γi : Ii → G, i = 1, 2 equivalent, denoted γ1 ∼ γ2, if (ϕoγ1)′(0) = (ϕoγ2)′(0).Clearly, this defines an equivalence relation. The equivalence classes are called tangentvectors at p. We write Tp(G) for the set of all tangent vectors at p and [γ] ∈ Tp(G) forthe equivalence class of the curve γ. For a Lie group G, we call T1(G) the Lie algebraof G and denote it by L(G).

For any smooth map of Lie groups such as ϕ : G1 → G2 and p ∈ G1, we obtain alinear map

Tp(ϕ) : Tp(G1)→ Tϕ(p)(G2), [γ] 7→ [ϕoγ]

that is called the differential of ϕ in p.∗ speaker

36

V. Atayi and R. Kamyabi-Gol

Proposition 2.1. If ϕ : G1 → G2 is a homomorphism of Lie groups, then its differentialin 1, namely,

L(ϕ) := T1(ϕ) : L(G1)→ L(G2),

is a homomorphism of Lie algebras. Moreover, (expG2 )oL(ϕ) = ϕo(expG1 ).

Let V be a finite-dimensional vector space. We write GL(V) for the group of alllinear automorphisms of V . It is well known that GL(V) carries a natural Lie groupstructure.For a Lie group G, a homomorphism ϕ : G → GL(V) is called a representation of G onV . Similarly, if g is a Lie algebra, then a homomorphism of Lie algebras ϕ : g→ gl(V)is called a representation of g on V , in which, by gl(V) we mean L(GL(V)) which isequal to End(V) the group of all linear endomorphisms of V .

Let g be a finite-dimensional Lie algebra and define

Aut(g) := g ∈ GL(g) : (∀x, y ∈ g)g[x.y] = [gx.gy],

by lemma 4.2.2 of [7] we deduce the Lie algebra of Aut(g) as:

aut(g) = L(Aut(g)) = D ∈ gl(g) : (∀x, y ∈ g)D[x.y] = [Dx.y] + [x.Dy].

The elements of this Lie algebra are called derivations of g, and aut(g) is also denotedby der(g).

3. Main results

Definition 3.1. Let g and h be Lie algebras and α : g → der(h) be a homomorphism.Then the direct sum g ⊕ h of the vector spaces g and h is a Lie algebra with respect tothe bracket

[(x, y), (x′, y′)] := ([x, x′], [y, y′] + α(x)y′ − α(x′)y).

This Lie algebra is called the semidirect sum with respect to α of g and h. It is denotedby g ⊕α h. If α = 0, then g ⊕α h is called the direct sum of g and h, and it is denoted byg ⊕ h.

Let G and H be Lie groups and τ : G → Aut(H) be a group homomorphismdefining a smooth action (g, h) 7→ τg(h) of G on H. Then the product manifold G × His a group with respect to the product

(g, h)(g′, h′) = (gg′, hτg(h′))

and the inversion(g, h)−1 = (g−1, τg−1 (h−1)).

Since multiplication and inversion are smooth, this group is a Lie group, called thesemidirect product of G and H with respect to τ. It is denoted by G ⋉τ H.

Let (R+ ⋉τ R) ⋉λ R2 be the shearlet group, in which, for a ∈ R+ and for(a, s) ∈ R+ ⋉τ R, τa : R → R and λ(a,s) : R2 → R2 are given by τa(s) =

√as and

37

Shearlet Lie group

λ(a,s)(t1, t2) = S sAa(t1, t2), in which, S s =

(1 s0 1

)and Aa =

(a 00√

a

)are, respectively,

shearing and anisotropic(parabolic) scaling matrices acting on plane. Now we areable to compute Lie algebra of the shearlet group:

Theorem 3.2. Lie algebra of the shearlet group is given by:

L((R+ ⋉τ R) ⋉λ R2) (R ⊕α R) ⊕β R2,

in which, α : R → R and β : R ⊕α R → gl2(R) are given by α(x) = 12 x and

β(x, y) =(x y0 x

2

).

[1] G.S. Alberti, F. De Mari, E. De Vito, L. Mantovani, Reproducing Subgroups of Sp(2;R). Part II:Admissible Vectors, Monatsh. Math., 173(3), (2014), 261-307.

[2] V. Atayi, R.A. Kamyabi-Gol, On the Characterization of Subrepresentations of Shearlet Group,accepted in Wavelets and Linear Algebra.

[3] S. Dahlke, G. Kutyniok, P. Maass, C. Sagiv, H.-G. Stark, G. Teschke, The uncertainty principleassociated with the continuous shearlet transform, Int. J. Wavelets. Multiresolut. Inf. Process, 6,(2008), 157-181.

[4] P. Grohs, Continuous shearlet tight frames, J. Fourier Anal. Appl., 17(3), (2011), 506-518.[5] K. Guo, D. Labate, W.-Q. Lim, Edge analysis and identification using the continuous shearlet

transform, Appl. Comput. Harmon. Anal., 27(1), (2009), 24-46.[6] K. Guo, D. Labate, Optimally sparse multidimensional representation using shearlets, SIAM J.

Math. Anal. 39, (2007), 298-318.[7] J. Hilgert, K.H. Neeb, Structure and Geometry of Lie Groups, Springer, (2012).[8] R.A. Kamyabi-Gol, V. Atayi, Abstract Shearlet Transform, accepted in Bull. Belg. Math. Soc.

Simon Stevin.[9] E. Nobari, S.M. Hosseini, A method for approximation of the exponential map in semidirect

product of matrix Lie groups and some applications, J. Comput. Appl. Math., 234, (2010), 305-315.

V. Atayi,Department of Mathematics,Ferdowsi University of Mashhad,Mashhad,Irane-mail: [email protected]

R. Kamyabi-Gol,Center of Excellence in Analysis on Algebraic Structures,Department of Mathematics,Ferdowsi University of Mashhad,Mashhad 91775-1159,Irane-mail: [email protected]

38



January 20–21, 2016

pseudoframe multiresolution structure on the space MN×N(C)-module L2(R,CN)

H. Azarmi∗, M. Janfada and R. Kamyabi-Gol

AbstractLet N be a positive number and L2(R,CN) be the Hilbert MN×N (C)- module space. The concept ofa generalized multiresolution structure (GMS) in L2(R,CN ) is discussed which is a generalization ofGMS in L2(R). Basically a GMS in L2(R,CN ) consists of an increasing sequence of closed subspaces ofL2(R,CN ) and a pseudoframe of translation type at each level. Also, the construction of affine frames forL2(R,CN ) based on a GMS is presented.

2010 Mathematics subject classification: Primary 42C15, Secondary 46H25.Keywords and phrases: HilbertA- module , pseudoframe, multiresolution structure, Affin frame..

1. Introduction

The theory of frame was introduced by Duffin and Schaeffer [3]in the early 1950s todeal with problems in nonharmonic Fourier series; cf. [10]. There has been renewedinterest in the subject related to its role in wavelet theory [2] . Frames provide auseful model to obtain signal decompositions in cases where redundancy, robusteness,oversampling, and irregular sampling play a role, e.g., [1]. The theory of multireso-lution analysis (MRA) has its roots in image and multiscale signal processing, and isconcerned with the decomposition of signal into subspaces of different resolutions. Ithas played a fundamental role in the development of wavelet theory [1, 9]. The theoryof frame for Hilbert modules was introduced by Frank and Larson in [4].

Let A be a unital C∗- algebra and J be a finite or countable index set. A sequencex j j∈J of elements in Hilbert A- module H is said to be a frame if there exist twoconstants C,D > 0 such that

C.⟨x, x⟩ ≤∑j∈J⟨x, x j⟩⟨x j, x⟩ ≤ D.⟨x, x⟩

for every x ∈ H .This is equivalent to x j j is a frame forH with bounds C and D if and only if

C∥x∥2 ≤ ∥∑

j

⟨x, x j⟩⟨x j, x⟩∥ ≤ D∥x∥2

∗ speaker

39

H. Azarmi, M. Janfada and R. Kamyabi-Gol

If x j j be a Parsval frame then the reconstruction formula x =∑

j⟨x, x j⟩x j holdsfor every x ∈ H [5].

In [6], the notion of generalized multiresolution structure (GMS) in L2(R) wasintroduced. Basically, the GMS consists of an increasing sequence of closed subspaceof L2(R), with a pseudoframe of translates at each level. Let τkϕk∈Z and τkϕ

∗k∈Z betwo sequences in L2(R). Let X be a closed subspace of L2(R). We say τkϕk∈Z formsa pseudoframe of translates for X with respect to τkϕ

∗k∈Z if

x =∑

k

⟨x, τkϕ∗⟩τkϕ, x ∈ X.

In a more general case, let X be a closed subspace of a separable Hilbert space H .Let xnn ⊂ H be a Bessel sequence with respect to X, and let x∗n ⊂ H be a Besselsequence inH . We say xnn is a pseudoframe for the subspace X (PFFS) with respectto x∗nn if

x =∑

k

⟨x, xk⟩x∗k, x ∈ X.

x∗n is called a dual pseudoframe (or PFFS-dual) to xnn for the subspace X, (see [7]and [8] for more details).

A generalized multiresolution structure (GMS) V j,Φ,Ψ j∈Z of L2(R) is an in-cresing sequece of closed linear subspaces V j ⊆ L2(R) and two elements Φ,Ψ ∈ L2(R)for which the following hold:

1. ∪ jV j = L2(R),∩ jV j = 0,2. f (.) ∈ V j if and only if f (2.) ∈ V j=1

3. f ∈ V0 implies that τk f ∈ V0, for all k ∈ Z,4. τkΦ : k ∈ Z is a frame for V0 with respect to τkΨ : k ∈ Z.

Let N be an integer and L2(R,CN) = f = ( f1, ..., fN)T : fi ∈ L2(R); i = 1, 2, ...,N.LetA = MN×N(C) be a C∗- algebra and also

A× L2(R,CN) −→ L2(R,CN), (A, f ) (A. f )(t) = A. f (t).

So L2(R,CN) is anA- Hilbert module. Now, we define an inner product by

⟨ f , g⟩ =∫R

f (t)h∗(t)dt ∈ MN×N(C)

note that "*" is conjugate and transpose. define ∥f∥2 = ∥⟨f, f⟩∥MN×N (C).

2. Definition and existence of pseudoframe for subspaces of L2(R,CN)

Now, we generalize the notion of GMS for L2(R,CN).

Definition 2.1. Let Φ,Ψ ∈ L2(R,CN) and X be a closed subspace of L2(R,CN). Thefamily τkΦk∈Z is said to be a pseudoframe with respect to τkΨk∈Z for X, if for every

40

pseudoframe multiresolution structure on the space MN×N (C)- module L2(R,CN )

f ∈ X,

f =

∑k∈Z

∫R

f1(t)ψ1(t − k)τkϕ1 + ... +∫R

f1(t)ψN(t − k)τkϕN]...∑

k∈Z∫R

fN(t)ψ1(t − k)τkϕ1 + ... +∫R

fN(t)ψN(t − k)τkϕN]

=

∑k∈Z⟨ f , τkΨ⟩τkΦ

It is important to note that τkΦ and τkΨ need not be contained in X. Also they arenot generally commutable, this means there exists f ∈ X such that the following is nottrue, f =

∑k∈Z⟨f, τkΦ⟩τkΨ.

In the following theorem, we are going to find a sufficient and necessary conditionfor the functions Φ and Ψ such that their translations forms a pseudoframe.

Theorem 2.2. Let Φ = (ϕ1, ., ., ., ϕN)T ∈ L2(R,CN) be such that for i = 1, ., ., .,N; |ϕi| >0, a.e. on a connected neihborhood of 0 in [− 1

2 ,12 ), and ϕi = 0, a.e. otherwise. For a

fixed c > 0, let ∆ := γ ∈ R : |ϕi(γ)| ≥ c, and let V0 := f ∈ L2(R,CN) : suppf ⊆ ∆.Now for a Ψ ∈ L2(R,CN), τkΦk∈Z forms a pseudoframe for V0 with respect toτkΨk∈Z if and only if (

ϕiψ j

)i, jχ∆ = χ∆; a.e. (1)

Moreover, if Ψ is also such that on [− 12 ,

12 ) , |ψi| > 0 and the above equality holds,

then τkΦ and τkΨ commute, in the sense that for any f ∈ X, f =∑

k∈Z⟨f, τkΨ⟩τkΦ =∑k∈Z⟨x, τkΦ⟩τkΨ.

3. Generalized Multiresolution Structure and Affine Pseudoframe

By applying Theorem 2.2, we are going to construct a generalized multiresolutionstructure for L2(R,CN).

Definition 3.1. A generalized multiresolution structure (GMS) V j,Φ,Ψ j∈Z of L2(R,CN)is an increasing sequece of closed linear subspaces V j ⊆ L2(R,CN) and two elementsΦ,Ψ ∈ L2(R,CN) for which the following hold:

1. ∪ jV j = L2(R,CN),∩ jV j = 0,2. f(.) ∈ V j if and only if f(2.) ∈ V j+1

3. f ∈ V0 implies that τkf ∈ V0, for all k ∈ Z,4. τkΦ : k ∈ Z is a frame for V0 with respect to τkΨ : k ∈ Z.

Proposition 3.2. Suppose that τkΦk∈Z is a pseudoframe for V0 with respect toτkΨk∈Z and V j := f ∈ L2(R,CN) : D− jf ∈ V0, then Φ jkk∈Z is a pseudoframefor V j with respect to Ψ jkk∈Z such that θ jk = 2

j2 θ(2 j. − k).

Theorem 3.3. Let Φ,Ψ ∈ L2(R,CN) and V0 has the properties specified in Theorem2.2 and V j is similar to Theorem 3.2, then V j,Φ,Ψ j forms a GMS for L2(R,CN).

41

H. Azarmi, M. Janfada and R. Kamyabi-Gol

Let two sequences of matrices h0(k)k and h∗0(k)k be such that the followingsummations are convergent, H0(.) =

∑n h0(n)e−2πin.,H∗0(.) =

∑n h∗0(n)e−2πin.. We say

H0 and H0∗ generate Φ and Ψ, respectively, if

Φ(t) =

√2∑

n(h110 (n)ϕ1(2t − n) + ... + h1N

0 (n)ϕN(2t − n))...√

2∑

n(hN10 (n)ϕ1(2t − n) + ... + hNN

0 (n)ϕN(2t − n))

(2)

and

Ψ(t) =

√2∑

n(h110∗(n)ψ1(2t − n) + ... + h1N

0∗ (n)ψN(2t − n))...√

2∑

n(hN10∗ (n)ψ1(2t − n) + ... + hNN

0∗ (n)ψN(2t − n))

(3)

.In terms of the filters H0,H0∗, Theorem 2.2 becomes the following:

Theorem 3.4. Suppose H0 and H0∗ generate Φ and Ψ as above. Assume Φ,Ψ ∈L2(R,CN) have the properties specified in Theorem 2.2. If τkϕk forms a pseudoframeof translations for V0 with respect to τkΨk then(∑N

i=1 Hni0 Hmi

0∗

)n,m

.χ ∆2= 2χ ∆

2, a.e. (4)

We shall denote the orthogonal complement of V0 in V1 by W0, as usual, in orderto split a function f of V1 into two functions in V0 and W0, respectively.

Definition 3.5. Let V j,Φ,Ψ j be a given GMS and Φ1,Ψ1 be two functions inL2(R,CN). We say τkΦ, τkΦ

1k∈Z is an affine pseudoframe for V1 with respect toτkΨ, τkΨ

1k∈Z, if and only if f =∑

k∈Z⟨f, τkΨ⟩τkΦ +∑

k∈Z⟨f, τkΨ1⟩τkΦ

1 In this caseτkΨ, τkΨ

1k∈Z is called a dual pseudoframe for τkΦ, τkΦ1k∈Z.

We are going to characterize conditions for which τkΦ, τkΦ1k∈Z is an affine

pseudoframe for V1 with respect to τkΨ, τkΨ1k∈Z. Let χ∆(γ) be IN×N on the interval

∆. We will also use the following 1-periodic function Λ∆(γ) ≡ ∑k χ∆(γ + k).

Theorem 3.6. Let ∆ be the bandwidth of the subspace V0 defined in Theorem 2.2.τnΦ, τnΦ

1n form a pseudoframe of translates for V1 with respect to τnΨ, τnΨ1n if

and only if there are G0 and G1 in L2(MN×N(T)) such that

G0(γ)H∗0∗(γ)Λ∆(γ) +G1H∗1∗(γ)Λ∆(γ) = 2Λ∆(γ) a.e,

G0(γ)H∗0∗(γ +12 )Λ∆(γ) +G1H∗1∗(γ +

12 )Λ∆(γ) = 0 a.e.

42

pseudoframe multiresolution structure on the space MN×N (C)- module L2(R,CN )

[1] J.J. Benedetto, Irregular Sampling and Frame, in Wavelets: A Tutorial in Theory and Applica-tions. (C. K. Chui, ED.), pp. 445- 507, academic Press, Boston, 1992.

[2] I. Daubechies, Ten Lecture on wavelets, SIM, Philadelphia, 1992.[3] R.Duffin and Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc72 (1952)

341-366.[4] M. Frank and D.R. Larson, Frame in Hilbert C∗-modules and C∗-algebras,J. Operator theory 48

( 2002), 273-314.[5] M. Frank and D.R. Larson, A module frame concept for Hilbert C∗-modules, Contemp.

Math247(1999) 207-233.[6] S. Li, A theory of generalized multiresolution structure and pseudoframes of translates,J Fourier

Anal and Appl( 2001).[7] S. Li and H. Ogawa, A theory of peseudoframes for subspaces with applications, Tech Report

(1998).[8] S. Li and H. Ogawa, Pseudoframes for subspaces with applications,J Fourier Anal Appl( 2004)

10: 409-431.[9] Y. Meyer, Waveletes and Operators, Cambridge Univ. Press, Cambridge, UK, 1992.

[10] R. Young, An Introduction to Nonharmonic Fourier series, Academic Press, New York, 1980.

H. Azarmi,Department of Mathematics,Ferdowsi University,City Mashhad, Irane-mail: azarmi_ [email protected]

M. Janfada,Department of Mathematics,Ferdowsi University,City Mashhad, Irane-mail: [email protected]

R. Kamyabi-Gol,Department of Mathematics,Ferdowsi University,City Mashhad, Irane-mail: [email protected]

43



January 20–21, 2016

Amenability and the compact multipliers

A. Bagheri salec∗ and M. Akbari Tootkaboni

Abstract

Let G be a locally compact group. There is a known theorem that prove amenability of the group algebraL1(G) is a necessary and sufficient condition to existance a compact or weakly compact right multiplieron L1(G)∗∗. In this paper focused on other algebras with this property. In this regard, we prove the abovesentence about weighted group algebras does not necessarily hold.

2010 Mathematics subject classification: Primary 43A22, Secondary 47B07.

Keywords and phrases: Amenable algebra, Compact operator, Group algebras.

1. Introduction

A Banach algebra A is said to be amenable if every continuous derivation D : A→ X∗

is inner for every Banach A-modul X. Also a group G is said to be amenable if thereis a positive translation-invariant functional m on L1(G) sach that m(1) = 1. Such afunction is called a mean on G. Johnson [6] showed that L1(G) is amenable if andonly if G is amenable. It is well known that if a Banach algebra A be amenable, thenA has a bounded approximate identity. Throughout this paper, G is a locally compactgroup with Haar measure λ. A weight on G is a continuous function ω : G → (0;+∞)with ω(xy) ≤ ω(x)ω(y), for all x, y ∈ G. The weighted group algebra is the algebraof measurable function on G which are integrable with respect to the weightω. That is,

L1(G, ω) = f : f is measurable and,∫

fωdλ < ∞.

In the following it is seen that amenability of the weighted group algebra L1(G, ω)is not necessarily equaivalevt to existance a compact or weakly compact multiplieron L1(G, ω)∗∗, (although by theorem 3.1 it is hold in the case ω = 1). Thereforeamenability of a banach algebra A is not equaivalent to existance a compact rightmultiplier on A∗∗ in general.

∗ speaker

44

A. Bagheri Salec andM. Akbari Tootkaboni

2. Preliminary and definitions

Corresponding to any weight ω another weight ω∗ is defined by ω∗(x) =ω(x)ω(x−1), for all x ∈ G, and the function Ω : G × G → (0,+∞), by Ω(x, y) =ω(xy)/ω(x)ω(y) for all x, y ∈ G. The extention of Johnson‘s theorem in weightedgroup algebras is the following theorem.

Theorem 2.1. The following statements are equivalent:(a) L1(G, ω) is amenable.(b) G is amenable and ω∗ is bounded.

Proof. See [5], theorem 0.

Definition 2.2. Let A be a Banach algebra. A bounded linear operator T : A → A iscalled a left (right) multiplier, if T (ab) = T (a)b(T (ab) = aT (b)), for all a, b ∈ A.

For every Banach algebra A, there exist two Arens products on the second dualA∗∗. If these two products coincide on A∗∗, then A is said to be Arens regular. It isknown that the group algebra L1(G) is Arens regular if and only if G is finite [9]. Theextention of this theorem in weighted group algebras is studied by Rejali and Vishkyin [7]. Recall that a weight ω on G is called zero cluster if,

limnlimmΩ(xn, xm) = 0 = limmlimnΩ(xn, xm).

for all sequence xn and xm in G with distinct element, whenever both iterated limitsare exists.

Theorem 2.3. The following statements are equivalent:(a) L1(G, ω) is Arens regular.(b) G is finite or G is disceret and Ω is zero cluster.(c) L1(G, ω)∗∗ is Arens regular.

Proof. Proof. See [7], theorem 2.

3. Main results

In the first we describe a known and very nice result.

Theorem 3.1. Suppose that G is a locally compact group. Then the following areequivalent:(a) G is ameanable;(b) There is a compact right multiplier T : L1(G)∗∗ → L1(G)∗∗;(c) There is a weakly compact right multiplier T : L1(G)∗∗ → L1(G)∗∗.

45

Amenability and the compact multipliers

Proof. See [3], theorem 3.1.

Therefore by theorem 3.1 and the Johnson theorem, amenability of the group al-gebra L1(G) is equaivalent to existence a compact or weakly compact operator onL1(G)∗∗. The natural question is:

Question 1. If A be a Banagh algebra with the bounded approximate identity, what isthe relation between amenability of A and existence of a compact, or weakly compactright multiplier on its second dual A∗∗? In other word is the theorem 3.1 remain validby replacement L1(G) by an arbitary Banach algebra with the bounded approximateidentity?

In the following we give a negative answer to the above question in weightedgroup algebras. Note that if L1(G, ω) is amenable then by [8] corollary 2, L1(G, ω)and L1(G) are isomorphic, and hence by theorem 3.1 there is a non-zero compact orweakly compact right multiplier on L1(G, ω).We need the following statements too.

Theorem 3.2. Suppose that G is a locally compact group. Then the following areequivalent:(a) G is compact(b) L1(G)∗∗ has a compact left multiplier T with < T (n), 1 >, 0 for some n ∈ L1(G)∗∗;(c) L1(G)∗∗ has a weaklycompact left multiplier T with < T (n), 1 >, 0 for somen ∈ L1(G)∗∗;

Proof. See [4], theorem 3.1.

Proposition 3.3. Let l1(G, ω) be commutative and Arens regular and l1(G, ω) has atopological left invariant mean, such as m with < m, 1 >, 0. Then l1(G, ω)∗∗ has aweakly compact left multiplier T , with < T (n), ω >, 0 (or < T (n), 1 >, 0), for somen ∈ l1(G, ω)∗∗.

Proof. Proof. See [2], proposision 1.

Example 3.4. Let α ∈ (0,+∞) and ωα : Z → (0,+∞), define by ωα(n) = (1 + n)α,(n ∈ Z). Then there is a weakly compact left multiplier T on l1(Z, ωα)∗∗.

Solution. For any commutative Banach algebra A, the second dual A∗∗ of A iscomutative under either Arens product, if and only if A is Arens regular, (see [1]).Since α is zero cluster, l1(Z, ωα) is Arens regular by theorem 2.3. Therefore, in thisexample l1(Z, ωα)∗∗ is commutative. On the other hand by proposition 3.3 there exista weakly compact left multiplier T on l1(Z, ωα)∗∗. Hence T is a weakly compact right

46

A. Bagheri Salec andM. Akbari Tootkaboni

multiplier on l1(Z, ωα)∗∗. Note that l1(Z, ωα) is not amenable by theorem 2.1, becouseω∗α is not bounded.

Remark 3.5. The example 3.4 shows that, we can not generalize theorem 3.1 to anyBanach algebra with the bounded approximate identity. We now have to answer thisquestion, that:Is there a suitable abstract status for theorem 3.1?

Acknowledgement

Acknowledgment The acuthors appreciate the efforts of the organizers of theconference.

[1] H. D. Dales, A. T.-M. Lau, The second duals of Beurling algebras, Mem. Amer. Math. 177 No.386 (2005).

[2] A. Bagheri Salec, A. Riazi, Multipliers on second dual algebras of weighted group algebras, Sci.Math. Japonica (2005), 527-529.

[3] F. Ghahramani, A. T.-M. Lau, Multipliers and ideals in second conjugate algebras related tolocally compact groups, J. Funct. Anal. 132 (1995).

[4] F. Ghahramani, A. T.-M. Lau, Multipliers and Modulus on Banach algebras related to locallycompact groups, J. Funct. Anal. 150 (1997), 478-497.

[5] N. Gronbaek, Amenability of weighted convolution algebras on locally compact groups, Trans.Amer. Math. Soc. 319 (1990), 765-775.

[6] B. E. Johnson, Cohomology in Banach algebras, Mem. Amer. Math. Soc. 127 (1972).[7] A. Rejali, H. R. E. Vishki, Regularity and amenability of the second dual of weighted group

algebras, Proyecciones 23 (2007), 259-267.[8] M. White, Characters on weighted amenable groups, Bull. London Math. Soc. 23 (1991), 375-380.[9] N. Young, The irregularity of multiplication in group algebras, Quart. J. Math. Oxford 24(2)

(1973), 59-62.

A. Bagheri salec,Department of Mathematics,University of Qom,City Qom, Irane-mail: alireza−bagheri−[email protected]

M. Akbari Tootkaboni,Department of Mathematics,University of Shahed,City Tehran, Irane-mail: [email protected]

47



January 20–21, 2016

Some Basic Theorems In Operator Valued Measure Theory

G. A. Bagheri-Bardi and M. Khosheghbal-Ghorabayi ∗

Abstract

Let Ω be a measurable space and B(H) be the set of all bounded linear operators on a Hilbert space Hwhich equipped with several σ-algebras. In this talk we present the extensions of Egoroff’s Theorem andLusin’s Theorem for measurable maps from Ω to B(H).

2010 Mathematics subject classification: Primary 47A56; Secondary 28A05, 28A20.Keywords and phrases: operator valued maps, measurability, σ-algebras, sequences of measurable maps.

1. Introduction

Let H be a Hilbert space and B(H) be the set of all bounded linear operators onH . There are some well-known locally convex vector topologies on B(H) which aregiven in the following diagram. For definitions and details see ([1][8][9][10]). Wejust recall the Arens-Mackey topology which is less classical than the six other ones.This locally convex vector topology is given as the uniform convergence topology onσ(B(H),B(H)∗)-compact convex subsets of B(H)∗.

Arens-Mackey ⊃ σ-strong∗ ⊃ σ-strong ⊃ σ-weak∪ ∪ ∪strong∗ ⊃ strong ⊃ weak

(1)

where ⊃means that the right-hand side is coarser than the left-hand side. The notationsτ and Mτ run just over these seven topologies and the σ-algebras generated by them,respectively. In the sequel, M is a σ-algebra on a non-empty set Ω and (Ω,M, µ) is ameasure space. A map φ : (Ω,M)→ (B(H),Mτ) is measurable, if for any measurableset v of Mτ, it is concluded φ−1(v) ∈ M.

The measurability of operator valued maps φ : (Ω, M) → (B(H), Mτ) was firstinvestigated by Johnson for special cases which τ is strong operator topology and His separable Hilbert space [5]. He showed the product of finite strong operator mea-surable maps is strong operator measurable when H is separable. Here, we show by∗ speaker

48

G. A. Bagheri-Bardi andM. Khosheghbal-Ghorabayi

discarding a measurable subset of Ω and a closed subspace of (not necessary separa-ble) Hilbert space, the pointwise-strongly convergency turns to uniformly convergency(operator valued Egoroff’s Theorem) as well as every operator valued measurable mapacts same as some τw-continuous operator valued map (operator valued Lusin’s theo-rem) when τ is one of the topologies mentioned in the Diagram (1).

2. Second section

Definition 2.1. Let φnn∈N and φ be operator valued maps on Ω. We say that φn ispointwise-strongly convergent to φ, denoted by φn

p.s−→ φ, if

∥(φn(x) − φ(x))ξ∥ → 0 (x ∈ Ω , ξ ∈ H).

Example 2.2. Suppose µ is the normalized arc length measure (Haar measure) on thecircle group T and for 1 ≤ p < ∞ set

Lp(T ) = f : T → C : f is measurable and∫| f (λ)|pdµ(λ) < ∞.

Thus, if f ∈ L1(T ), then∫

f (λ)dµ(λ) = 12π

∫ 2π0 f (eit)dt.

For each integer n ∈ N, the function εn : T → T, λ → λn, is of course continuousand εnn∈N is an orthonormal basis for Hilbert space L2(T ). Denote by B(L2(T )), theset of all bounded linear operators on L2(T ). For each t in the circle group T , thereexists a unique θt ∈ [0, 2π) such that t = eiθt . Take

Et = λ ∈ T : 0 ≤ θλ ≤ θt,

and for every n ∈ N, set

Etn = λ ∈ T : 0 ≤ θλ ≤ (1 − 1

n)θt.

Consider the operators φ(t) and φn(t) in B(L2(T )) as follows:

φ(t)( f ) = χEt f , φn(t)( f ) = χ

Etn

f ( f ∈ L2(T )).

Now define the following maps

φ : T → B(L2(T )) ; t → φ(t),

andφn : T → B(L2(T )) ; t → φn(t).

Thus φn is pointwise-strongly convergent to φ.

At the beginning, we recall the classic Egoroff’s Theorem [3] and the Noncommu-tative Egoroff’s Theorem [8] :

49


Theorem 2.3. (Classic Egoroff’s Theorem) Let (Ω,M, µ) be a finite measure spaceand fn be a sequence of measurable functions on Ω such that fn → f a.e. Thenfor every ϵ > 0 , there exists a measurable subset E of Ω with µ(E) < ϵ such that fnconverging uniformly to f on Ec

Theorem 2.4. ( Noncommutative Egoroff’s Theorem) Let A be a von Neumannalgebra. Let S be a bounded subset of A and S be its strong closure. Take an arbitraryelement a ∈ S . Then for any positive φ ∈ A∗ and any ϵ > 0, there exist a projectionp ∈ A and a sequence an in S such that

∥(an − a)p∥ → 0 , φ(1 − p) < ϵ

Furthermore, it is proved in [2] that the bonded assumption in the NoncommutativeEgoroff’s Theorem [8] is redundant. Now we present the following one:

Theorem 2.5. (Operator valued Egoroff’s Theorem) Suppose (Ω, M, µ) is a finitemeasure space . Let φn : (Ω,M) → (B(H),Mτ)n∈N be a sequence of measurablemaps converging pointwise-strongly to an operator valued measurable map φ. Thenfor an arbitrary positive element ω ∈ L1(H) and ϵ > 0 and δ > 0, there exists ameasurable set E ⊆ Ω and a projection p ∈ B(H) satisfying µ(E) < ϵ and ω(p) < δ,such that

supx∈Ec ∥(φn(x) − φ(x))(1 − p)∥ → 0.

Proof. To prove this, first we show the following steps:step 1: Mw, the σ-algebra generated by weak operator topology, contains all basicelement of strong operator topology.step 2: For all locally convex topologies, τ, mentioned in the Diagram (1), the set ofall operator valued measurable maps from (Ω,M) to (B(H),Mτ) is a vector space.step 3: For ξ ∈ H and n ∈ N, we define the function

gn : Ω→ C ; x→ ∥(φn(x) − φ(x))ξ∥.

which is measurable for all topologies τ mentioned in Diagram (1).By using above matters and classic Egoroff’s Theorem and knowing that by TheoremII.1.6 of [8], there are some sequence αii∈N ∈ ℓ1

+ and an orthonormal set eii∈N ⊆ Hsuch that

ω =

∞∑i=1

αiei ⊗ ei.

we prove the result.

3. Third Section

Before we present our last Theorem, we remind the classic Lusin’s Theorem [6]and the Noncommutative Lusin’s Theorem [8].

50

G. A. Bagheri-Bardi andM. Khosheghbal-Ghorabayi

Theorem 3.1. (Classic Lusin’s Theorem) Let Ω be a locally compact and Haussdorffspace and (Ω,M, µ) be a finite measure space and f be any measurable function onΩ. Then for every ϵ > 0 , there exists a measurable subset E of Ω with µ(E) < ϵ and afunction g ∈ Cc(Ω) s.t f = g on Ec.

Theorem 3.2. (Noncommutative Lusin’s Theorem) Let B be a C∗-algebra of a vonNumann algebra A and φ ∈ A∗ be positive. Then for every ϵ > 0 and any a ∈ A, thereexists a projection p ∈ A and b ∈ B such that

ap = bp , φ(1 − p) < ϵ

Now we imply the following one:

Theorem 3.3. (Operator valued Lusin’s Theorem) Let Ω be a locally compact andHaussdorff space and (Ω,M, µ) be a finite measure space and φ : (Ω,M, µ) →(B(H),Mτ) be any measurable map. Then for every ω ∈ L1(H)+ and arbitraryϵ > 0 , δ > 0, there exist a measurable set E ⊆ Ω with µ(E) < ϵ and a projectionp ∈ B(H) with ω(p) < δ and a map g ∈ Cc(Ω, (B(H), τw)) such that

(1 − p)( f (x) − g(x))(1 − p) = 0 (x ∈ Ec).

which τw is the weak topology on B(H).

Proof. For ξ, η ∈ H, we define the complex valued measurable function

φξ,η : (Ω,M, µ)→ C ; x→ ⟨φ(x)ξ, η⟩.

Then we prove our claim by using Theorem II.1.6 of [8] and applying the classicLusin’s Theorem for special type of above measurable functions.

[1] C. A. Akemann, The dual space of an operator algebra, rans. Amer. Math. Soc. 126 (1967) 749-752.

[2] C. A. Akemann, G. A. Bagheri-Bardi, A stronger noncommutative Egoroff’s theorem, NihonkaiMath. J. 25 (2014) 65-68.

[3] C. D. Aliprantis and O. Burkinshaw, Principles of real analysis, 3rd edition, Academic Name,1998.

[4] G. A. Bagheri-Bardi, Operator-valued measurable functions, Bull. Belg. Math.l Soc. Simon Stevin22 (2015) 159-163.

[5] G. W. Johnson, The product of strong operator measurable functions is strong operator measur-able, Proc. Amer. math. Soc. 4 (1993) 1097-1104.

[6] W. Rudin, Real and complex analysis, 3rd edition, McGraw Hill, 1986.[7] G. J. Murphy, C*-algebra and operator theory, Academic press, 1990.[8] M. Takesaki, Theory of operator algebra I, Springer-Verlag, New York, 1979.[9] J. Von Neumann, Zur Algebra der Funktionalopration und Theorie der normalen Operatoren,

Math. Ann. 102 (1929) 370-427.[10] J. Von Neumann, On a certain topology for rings of opeartors, Math. Ann. 37 (1936) 111-115.

51


G. A. Bagheri-Bardi,Department of Mathematics,Persian Gulf University,Boushehr 75168, Irane-mail: [email protected], [email protected]

M. Khosheghbal-Ghorabayi ,Department of Mathematics,Persian Gulf University,Boushehr 75168, Irane-mail: [email protected], [email protected]

52



January 20–21, 2016

The Stone -Cech Compactification of Groupoids

F. Behrouzi

AbstractLet G be a discrete groupoid and consider the Stone-Cech compactification βG of G. We will extend theoperation on the set of composable elements G(2) of G to an operation ′′∗′′ on a subset (βG)(2) of βG× βGsuch that the triple (βG, (βG)(2), ∗) is compact right topological semigroupoid.

2010 Mathematics subject classification: Primary 47A15; Secondary 46A32, 47D20.Keywords and phrases: Grouoids; Banach algebra; Stone -Cech compactification.

1. Introduction

A compactification of a topological space X is a compact space K together with anembedding e : X −→ K with e(X) dense in K. We usually identify X with e(X) andconsider X as a subspace of K. There exists a very special type of compactificationof X in which X is embedded in such a way that every bounded, real-valued (complex-valued) continuous function on X will extend continuously to the compactification.Such a compactification of X is called the Stone-Cech compactification and denotedby βX.

As known, the Stone-Cech compactification βG of an infinite discrete group G canbe turned into a (compact) semigroup by an operation, extended from G [1, 2, 4]. Inthis paper, we deal with groupoids instead of groups. Unlike groups, in a groupoid G,the product is not defined for each two elements of G. But, the product defined on asubset of G × G, the set of composable pairs. The product on composable elementsis associative (see Definition 2.1 below ). We will show that, like the group case,the operation of any groupoid G can be extend to βG such that this operation is stillassociative.

2. Basics on groupoids

Here is some elementary definitions in groupoid literatures. For more details werefer the reader to [5–7].

Definition 2.1. A groupoid is a set G endowed with a product map (g, h) 7→ gh :G(2) −→ G where G(2) is a subset of G × G called the set of composable pairs and aninverse map g 7→ g−1: G → G such that the following relation are satisfied:

53

F. Behrouzi

1. (g−1)−1 = g.2. If (g, h) ∈ G(2) and (h, k) ∈ G(2), then (gh, k), (g, hk) ∈ G(2) and we have

(gh)k = g(hk).

3. (g−1, g) ∈ G(2) and if (g, h) ∈ G(2) then g−1(gh) = h.4. (g, g−1) ∈ G(2) and if (h, g) ∈ G(2) then (hg)g−1 = h.

The unit space G0 is the subset of elements gg−1 where g ranges over G. The rangmap r : G −→ G0 and the source map d : G −→ G0 is defined by r(g) = gg−1 andd(g) = g−1g. The pair (g, h) belongs to the set G(2) if and only if d(g) = r(h). For eachu ∈ G0, the subsets Gu and Gu are given by Gu = d−1(u), Gu = r−1(u).

Definition 2.2. A topological groupoid consists of a groupoid G and a topologycompatible with the groupoid structure :

1. (x, y) 7→ xy : G(2) −→ G is continuous where G(2) has the induced topology fromG ×G.

2. g 7→ g−1 : G −→ G is continuous.

If G is a topological groupoid, then the maps r, d are continuous. In addition, if G0

is Hausdorff in the relative topology, then G(2) is closed in G ×G.

3. discrete groupoids

Let G be a groupoid and g ∈ G. For any f ∈ B(G), we define the left g-translationand the right g-translation of f , respectively, by

Lg f (x) =

f (gx) x ∈ Gd(g)

0 x < Gd(g), Rg f (x) =

f (xg) x ∈ Gr(g)

0 x < Gr(g).

Since f is bounded, so are Lg f and Rg f . Therefore, for any θ ∈ βG and f ∈ B(G), wecan consider two new functions

Tθ, f : G −→ C , S θ, f : G −→ C

given byTθ, f (g) = θ(Lg f ) , S θ, f (g) = θ(Rg f ).

It is clear that Tθ, f and S θ, f are bounded. Next, we collect some elementary propertiesof these functions.

Lemma 3.1. Let f be a bounded function on a groupoid G. Then for any g, h ∈ G andany θ ∈ βG:

1. If (g, h) ∈ G(2) then Lh(Lg f ) = Lgh f . Also, if (g, h) < G(2) then Lh(Lg f ) = 0.2. Lg(Tθ, f ) = Tθ,Lg f and Rg(S θ, f ) = S θ,Rg f .

54


According to G, we define the set of composable elements of βG by

(βG)(2) =∪u∈G0

Gu ×Gu =∪u∈G0

βGu × βGu.

It is trivial that G(2) ⊆ (βG)(2). In the following result, we extend the operation ofG(2) to (βG)(2).

Theorem 3.2. Let G be a discrete groupoid. There is a unique operation ∗ on (βG)(2)

satisfying the following conditions:

1. For every (g, h) ∈ G(2), g ∗ h = gh.2. For every u ∈ G0 and g ∈ Gu, the map η 7→ g ∗ η : Gu −→ βG, is continuous.3. For every u ∈ G0 and η ∈ Gu, the map θ 7→ θ ∗ η : Gu −→ βG, is continuous.

Theorem 3.3. Suppose that G is a discrete groupoid and (θ, η) ∈ (βG)(2).

1. If gii∈I and h j j∈J are nets in G such that limi gi = θ and lim j h j = η, thenθ ∗ η = limi lim j gih j.

2. θ ∗ η( f ) = θ(Tη, f ).

One can consider the inversion map defined by g 7→ g−1 : G −→ G. By S2, thismap has a continuous extension inv : βG −→ βG. We denote again the inv(θ) byθ−1. By the continuity, if gii∈I is any net in G converging to θ in βG, then g−1

i i∈I

converges to θ−1. Consequently, (θ−1)−1 = θ and if θ ∈ Gu, then θ−1 ∈ Gu. Letf ∈ B(G) and define the transformation f on G by f (g) = f (g−1). This relation canbe extended to βG, that is, for any θ ∈ βG, we have θ−1( f ) = θ( f ). If G is a groupoid,then (g, g−1) ∈ G(2) for all g ∈ G. But this property does not hold for the Stone-Cechcompactification βG, unless Go is finite.

Theorem 3.4. Let G be a discrete groupoid. For every θ ∈ βG, (θ, θ−1) ∈ (βG)(2) if andonly if G0 is finite.

Lemma 3.5. Let G be a discrete groupoid, (θ, η) ∈ (βG)(2) and let v ∈ G0. Then

1. η ∈ Gv if and only if θ ∗ η ∈ Gv.2. θ ∈ Gv if and only if θ ∗ η ∈ Gv.

Definition 3.6. A semigroupoid is a triple (Λ,Λ(2), ∗) such that Λ is a set, Λ(2) is asubset of Λ × Λ, and

∗ : Λ(2) −→ Λ

is an operation which is associative in the following sense: if f , g, h ∈ Λ are such thateither

(i) ( f , g) ∈ Λ(2) and (g, h) ∈ Λ(2), or(ii) ( f , g) ∈ Λ(2) and ( f ∗ g, h) ∈ Λ(2), or(iii) (g, h) ∈ Λ(2) and ( f , g ∗ h) ∈ Λ(2)

55

F. Behrouzi

then all ( f , g), (g, h), ( f ∗ g, h) and ( f , g ∗ h) lie in Λ(2), and

( f ∗ g) ∗ h = f ∗ (g ∗ h).

Moreover, for f ∈ Λ, we will set

Λ f = g ∈ Λ : ( f , g) ∈ Λ(2) , Λ f = g ∈ Λ : (g, f ) ∈ Λ(2).

Let (Λ,Λ(2), ∗) and (Λ′,Λ′(2), ∗′) be semigroupoids. A map T : Λ −→ Λ′ is calledhomomorphism if ( f , g) ∈ Λ(2), then (T ( f ),T (g)) ∈ Λ′(2) and T ( f ∗ g) = T ( f ) ∗′ T (g).

Definition 3.7. Let (Λ,Λ(2), ∗) be a semigroupoid and a topological space. Then(i) Λ is called left topological semigroupoid if for every f ∈ Λ the map g 7→ f ∗ g :

Λ f −→ Λ is continuous.(ii) Λ is called right topological semigroupoid if for every f ∈ Λ the map

g 7→ g ∗ f : Λ f −→ Λ is continuous

Let Λ be a right topological semigroupoid. The topological center of Λ is the setof all f ∈ Λ such that the map g 7→ f ∗ g : Λg −→ Λ is continuous.

Theorem 3.8. If G is discrete groupoid, then (βG, (βG)(2), ∗) is a compact righttopological semigroupoid. Moreover, the topological center of βG contains G.

Theorem 3.9. Let G be a discrete groupoid and let (K,K(2), ⋆) be a compact righttopological semigroupoid which is such that the following properties are satisfied:1. there is a morphism e : G → K such that e(G) is dense in K.2. the topological center of K contains e(G).3.

∪u∈G0 e(Gu) × e(Gu) ⊆ K(2).

Then there exists a continuous surjective homomorphism T : βG −→ K such that foreach g ∈ G, T (g) = e(g).

4. Topological Groupoids

Let G be a topological groupoid such that for every u ∈ G0, Gu is C*-embedded.Let lug be the extension of the map lug mentioned in the previous section. Then foreach fixed η ∈ Gu, we may consider the mapping ru

η from Gu into βG. defined byruη(g) = lug(g). But unlike the discrete case, nothing guarantees that the mapping ru

η iscontinuous for every η ∈ Gu. Therefore we might not able to extend these mappingsto βG leading to a continuous operation on βG.

We can start this process by extending the mappings ruh : Gu −→ βG to mappings

ruh : Gu −→ βG where h ∈ Gu. If for every θ ∈ Gu we define luθ : Gu −→ βG

by luθ(h) = ruh(θ), again nothing guarantees the continuity of the mappings lθ for each

θ ∈ Gu.Let G be a topological groupoid. As the previous section, we can define left g-

translation Lg f and right g-translation. But these map are not continuous in general.

56


As we know we can regard βG as the maximal ideal space of Cb(G). It seems that wecan not define the extended operation on βG in the term of elements of the maximalideal space of Cb(G). Because the mapping g 7→ η(Lg f ) is not well-defined.

Lemma 4.1. Suppose that u ∈ G0 , η ∈ Gu and f ∈ Cb(Gu). Let F1, F2 ∈ Cb(G)suchthat F1|Gu = F2|Gu = f . Then η(F1) = η(F2).

Let u ∈ G0 , g ∈ Gu and let h ∈ Gu. For f ∈ Cb(G) define Lug f : Gu −→ C by

Lug f (x) = f (gx). Also, define Ru

h f : Gu −→ C by Ruh f (x) = f (xh).

Definition 4.2. Let u ∈ G0, η ∈ Gu, θ ∈ Gu. For f ∈ Cb(G), set

T uη, f : Gu −→ C T u

η, f (g) = η(Lug f ).

where Lug f is an extension of Lu

g f in Cb(G). By lemma 4.1, T uη, f is well-defined. Also

defineSuθ, f : Gu −→ C Su

θ, f (h) = θ(Ruh f ).

where Ruh f is an extension of Ru

h f in Cb(G).

Theorem 4.3. Let G be a topological groupoid and let f ∈ Cb(G). Then the followingsare equivalent:1. For every u ∈ G0 and for every θ ∈ Gu, luθ is continuous.2. For every u ∈ G0 and for every η ∈ Gu, ru

η is continuous.3. For every u ∈ G0, for every η ∈ Gu and every f ∈ Cb(G), T u

h f is continuous.4. For every u ∈ G0, for every θ ∈ Gu and every f ∈ Cb(G), Su

θ, f is continuous.

Example 4.4. Let G be the groupoid [0,∞)×[0,∞). Consider the sequence ((1, n))∞n=1.So there exist a subsequence ((1, nk))∞k=1 and η ∈ βG such that (1, nk) −→ η in βG.Let f be a function in Cb([0,∞)) such that f (n) = 1 for every n and f (t) = 0 ifn + 1

n < t < n + 1 − 1n+1 . Define F : G −→ C by F(x, y) = f (x + y). Then for every

m ∈ N, Tη,F( 1m , 1) = limk f (nk +

1m ) = 0. But Tη,F(0, 1) = f (nk) = 1. Therefore, Tη,F

is not continuous.

[1] M. Filali and I. Protasov. Ultrafilters and topologies on Groups. Math. Stud. Monogr. Ser., VNTL,Lviv, 2006.

[2] M. Filali and T. Vedenjuoksu, The Stone- -Cech compactication of a topological group and theβ-extension property, Houston J. Math. 36 no. 2 (2010) 477-488.

[3] L. Gillman and M. Jerison, Rings of continuous functions, van Nostrand, Princeton, 1960.[4] N. Hindman and D. Strauss. Algebra in the Stone-Cech compactification theory and applications.

Water de Gruyter, Berlin, 1998.[5] P. Muhly, Coordinates in operator algebra, (Book in preparation).[6] A. L. T. Paterson. Groupoids, inverse semigroups, and their operator algebras, volume 170 of

Progress in Mathematics. Birkhäuser Boston Inc., Boston, MA, 1999.[7] J. Renault, A groupoid approach to C∗-algebras, Lecture Notes in Math. Springer-Verlag, 793,

1980.

57

F. Behrouzi

[8] R. Walker, The Stone-Cech compactification, Springer-Verlag, Berlin, 1974.

F. Behrouzi,Department of Mathematics,University of Alzahra,Tehran , Irane-mail: [email protected]

58



January 20–21, 2016

Separating ideals in the algebra of uniformly continuousintegrable functions

M. Dashti∗ and R. Nasr-Isfahani

Abstract

Let G be a locally compact group. We define a norm on the set L1(G) ∩ LUC(G) to make it a Banachspace and show that it is a semisimple Banach algebra. We also show that if G is a noncompact locallycompact group, then L1(G) ∩ LUC(G) contains no nontrivial separating ideal.

Keywords and phrases: Banach algebra, group algebra, uniformly continuous functions.2010 Mathematics subject classification: 43A20, 46A05.

1. Introduction

Let G be a locally compact group with a fixed left Haar measure λ. Let L1(G) denotethe space of all Haar integrable functions on G which is a Banach algebra with respectto the norm ∥.∥1 and the convolution product ∗ given by

(ϕ ∗ ψ)(x) =∫

Gϕ(xy)ψ(y−1)dy

for all ϕ, ψ ∈ L1(G) and almost all x ∈ G. The space Cb(G) of all bounded continuousfunctions on G is a Banach algebra with respect to the poinwise product and the norm

∥ f ∥sup = sup| f (x)| : x ∈ G

for all f ∈ Cb(G). We denote Cc(G) the subspace of Cb(G) consisting of functions withcompact support. For a function f on G and x ∈ G, the left translation Lx f is definedby Lx f (y) = f (x−1y) for all y ∈ G. Let LUC(G) denote the space of bounded leftuniformly continuous functions on G; i.e., all f ∈ Cb(G) such that the map x 7→ Lx fof G into Cb(G) is norm continuous; see [3] for more details.

A dense subspace S 1(G) of L1(G) is said to be a Segal algebra on G if it satisfiesthe following conditions.

(1) S 1(G) is a Banach space under some norm ∥.∥s, and for each f ∈ S 1(G) wehave

∥ f ∥1 ≤ ∥ f ∥s.∗ speaker

59

M. Dashti and R. Nasr-Isfahani

(2) S 1(G) is left transition invariant and the map x 7→ Lx f of G into S 1(G) iscontinuous for all f ∈ S 1(G).

(3) For each f ∈ S 1(G) and x ∈ G, we have

∥Lx f ∥s = ∥ f ∥s.

We denote the Jacobson radical of an algebraA by rad(A) that is the intersectionof the maximal modular left ideals ofA. The algebraA is semisimple if rad(A) = 0.For an ideal I inA, rad(I) = I ∩ rad(A); see [1], Section 1.5.

Recall that a closed ideal I in an algebra A is called a separating ideal if foreach sequence (an) in A there exists n0 ∈ N such that (a1...anI) = (a1...an0 I) and(Ia1...an) = (Ia1...an0 ) for all n ≥ n0; see [1], Section 5.2.

In this paper, we first show L1(G) ∩ LUC(G) is a semisimple Banach algebra andthen in the sequel, we shall prove that for a noncompact locally compact group G thereis not a nonzero finite dimensional ideal in L1(G) ∩ LUC(G). We shall use this resultand prove that for a noncompact locally compact group G, the only separating ideal inL1(G) ∩ LUC(G) is zero.

2. L1(G) ∩ LUC(G) as a Banach algebra

Let G be a locally compact group. In this section, we investigate the spaceL1(G) ∩ LUC(G) with norm ∥|.∥| = ∥.∥1 + ∥.∥sup. We first have the following result.

Proposition 2.1. Let G be a locally compact group. The Banach space L1(G)∩LUC(G)with convolution product is a Segal algebra. Consequently L1(G) ∩ LUC(G) is a leftideal of L1(G) and for each f ∈ L1(G)∩ LUC(G) and ϕ ∈ L1(G), ∥|ϕ ∗ f ∥| ≤ ∥ϕ∥1∥| f ∥|.In particular L1(G) ∩ LUC(G) is a Banach algebra under ∥|.∥|.

Proof. Using this fact L1(G) ∗ LUC(G) = LUC(G), we get L1(G) ∩ LUC(G) is a leftideal in L1(G). Since Cc(G) ⊆ L1(G) ∩ LUC(G), it follows that L1(G) ∩ LUC(G)is a dense subspace of L1(G). Clearly, L1(G) ∩ LUC(G) is translation invariant and∥|Lx f ∥| = ∥| f ∥| for all f ∈ L1(G)∩LUC(G) and x ∈ G. Let f ∈ L1(G)∩LUC(G) and letxα → e in G. Then ∥Lxα f − f ∥1 → 0 and ∥Lxα f − f ∥sup → 0. Hence ∥|Lxα f − f ∥| → 0.So, the map x 7→ Lx f of G into L1(G) ∩ LUC(G) is continuous. To this end, letf ∈ L1(G) ∩ LUC(G) and ϕ ∈ L1(G). Then

∥|ϕ ∗ f ∥| = ∥ϕ ∗ f ∥1 + ∥ϕ ∗ f ∥sup

≤ ∥ϕ∥1∥ f ∥1 + ∥ϕ∥1∥ f ∥sup

≤ ∥ϕ∥1(∥ f ∥1 + ∥ f ∥sup)≤ ∥ϕ∥1 ∥| f ∥|

and the proof is complete.

Corollary 2.2. Let G be a locally compact group. Then L1(G) ∩ LUC(G) is asemisimple algebra.

60

Separating ideals in the algebra of uniformly continuous integrable functions

Proof. L1(G) is always semisimple algebra; see [1] Corollary 3.3.35. Since L1(G) ∩LUC(G) is a Segal algebra by Proposition 2.1, it follows that L1(G) ∩ LUC(G) is alsosemisimple; see for example [4], Theorem 3.3.

The following key result characterizes left translation invariant ideal in the Banachalgebra L1(G) ∩ LUC(G). First let us recall that an ideal I in L1(G) ∩ LUC(G) is lefttranslation invariant if Lx f ∈ I for every x ∈ G and f ∈ I.

Lemma 2.3. Let G be a locally compact group. Then every closed ideal in L1(G) ∩LUC(G) is left translation invariant.

Proof. Let f ∈ I and x ∈ G. Then the map s 7→ Ls f from G into L1(G) is continuous.Thus for every ϵ > 0, there exists a symmetric compact neighborhood V1 of e in Gsuch that

∥ f − Ls f ∥1 = ∥Lx f − Lx(Ls f )∥1 ≤ϵ

2for all s ∈ V1. Since f ∈ LUC(G), it follows that for every ϵ > 0, there exists asymmetric compact neighborhood V2 of e in G such that

|Lx f (y) − Lx(Ls f )(y)| ≤ ϵ

2for all s ∈ V2 and y ∈ G. Set V := V1 ∩ V2. Then

∥Lx f − λ(V)−1(LxχV ∗ f )∥1 =

∫Gλ(V)−1

∣∣∣∣∣∫V

f (x−1y)ds −∫

Vf (s−1x−1y)ds

∣∣∣∣∣ dy

≤ λ(V)−1∫

V

(∫G| f (x−1y) − f (s−1x−1y)|dy

)ds

= λ(V)−1∫

V∥Lx f − Lx(Ls f )∥1ds

≤ sups∈V∥Lx f − Lx(Ls f )∥1.

In particular, for each y ∈ G we have,

|Lx f (y) − λ(V)−1(LxχV ∗ f )(y)| = λ(V)−1∣∣∣∣∣∫

Vf (x−1y)ds −

∫V

f (s−1x−1y)ds∣∣∣∣∣

≤ λ(V)−1∫

V| f (x−1y) − f (s−1x−1y)|ds

= λ(V)−1∫

V|Lx f (y) − Lx(Ls f )(y)|ds

≤ ∥Lx f − Lx(Ls f )∥sup.

Therefore

∥|Lx f−λ(V)−1(LxχV∗ f )∥| = ∥Lx f−λ(V)−1(LxχV ∗ f )∥1+∥Lx f−λ(V)−1(LxχV∗ f )∥sup < ϵ.

As I is a closed ideal, it follows that Lx f ∈ I.

61

M. Dashti and R. Nasr-Isfahani

Proposition 2.4. Let G be a locally compact group. Suppose that there is a nonzerofinite dimensional ideal in L1(G) ∩ LUC(G). Then G is compact.

Proof. Assume towards a contradiction that G is noncompact. Let I be a nonzerofinite dimensional ideal in L1(G) ∩ LUC(G). Take ϕ ∈ I with ∥|ϕ∥| = 1. By theprevious Lemma, I is left translation invariant. Let S be a linear space generated byLxϕ : x ∈ G. Then S is finite dimensional. So, there exist elements x1, ..., xn ∈ Gsuch that Lx1ϕ, ..., Lxnϕ is a basis for S . Since all norms on S are equivalent, it followsthat there exists a constant M ≥ 1 such that

n∑i=1

|βi| ≤ M∥|ψ∥|

for all ψ =∑n

i=1 βiLxiϕ ∈ S with β1, ..., βn ∈ C. Choose ϵ > 0 such that (M+1)ϵ < ∥ϕ∥1.Then there is a compact subset K of G such that

∫G\K |ϕ|dλ < ϵ; that is,

∫K |ϕ|dλ >

∥ϕ∥1 − ϵ. Thus for each x ∈ G, ∫xK|Lxϕ|dλ > ∥ϕ∥1 − ϵ.

Since G is noncompact, there exists x ∈ G such that xK is disjoint from x1K∪ ...∪ xnK.Now, suppose that Lxϕ =

∑ni=1 αiLxiϕ, where α1, ..., αn ∈ C. Then∫

xK|Lxϕ|dλ ≤

n∑i=1

|αi|∫

xK|Lxiϕ|dλ < Mϵ

and so ∥ϕ∥1 < (M + 1)ϵ, a contradiction.

The following main result is an immediate consequence of Corollary 2.2 andProposition 2.4.

Theorem 2.5. Let G be a noncompact locally compact group. Then the only separatingideal in L1(G) ∩ LUC(G) is 0. Consequently, every epimorphism onto L1(G) ∩LUC(G) and every derivation on L1(G) ∩ LUC(G) is continuous.

Proof. Let I be a nonzero separating ideal in L1(G) ∩ LUC(G), then I must be finitedimensional by Corollary 2.2 and Lemma 5.2.25 in [1]. So, G is a compact group byproposition 2.4, and this contradiction completes proof.

[1] H. G. Dales, Banach agebras and automatic continuity, London Mathematical Society Mono-graphs. New Series 24, Oxford Science Publications, The Clarendon Press, New York, 2000.

[2] Extremera, J., and Villena A. R., Separating Ideals in group algebras, Quart. J. Math. 55 (2004)303-305.

[3] E. Hewitt and B. Ross, Abstract harmonic analysis I, Springer-Verlag, New York, 1970.[4] M. Leinert, A contribution to Segal algebras, Manuscribta Math. 10 (1973) 297-306.

62

Separating ideals in the algebra of uniformly continuous integrable functions

M. Dashti,Department of Mathematical Sciences and Statistics,Malayer University,Malayer, Iran.e-mail: [email protected]

R. Nasr-Isfahani,Department of Mathematical Sciences,Isfahan University of Technology,Isfahan, Iran.e-mail: [email protected]

e-mail: [email protected]: [email protected]

63



January 20–21, 2016

Module mean for Banach Algebras

H. Ebrahimi∗ and A. Bodaghi

Abstract

In this paper, the module (ϕ, φ)-amenable Banach algebras are characterized. Also, the relations ofmodule (ϕ, φ)-amenability of a Banach algebra and their ideals are studied. Some mild conditions arefound for a Banach algebra to possess a module (ϕ, φ)-mean of norm 1.

2010 Mathematics subject classification: Primary 46H25; Secondary 43A07.Keywords and phrases: Banach modules; Module character amenability; Module mean. .

1. Introduction

For a non-zero character φ on a Banach algebra A, Kaniuth, Lau and Pym [11]introduced and studied the interesting notion of φ-amenability; see also [9, 13].Precisely, a Banach algebraA is φ-amenable if there exists a bounded linear functionalm on the dual spaceA∗ such that m(φ) = 1 and m( f · a) = φ(a)m( f ) for all a ∈ A andf ∈ A∗. Bodaghi and Amini [5] introduced the notion of module (ϕ, φ)-amenability fora class of Banach algebras that are modules over another Banach algebra as follows:

Let A and A be Banach algebras such that A is a Banach A-bimodule withcompatible actions, that is

α · (ab) = (α · a)b, (ab) · α = a(b · α) (a, b ∈ A, α ∈ A).Let ΦA be the character space of A and φ ∈ ΦA ∪ 0. Consider the linear mapϕ : A −→ A such that

ϕ(ab) = ϕ(a)ϕ(b), ϕ(a · α) = ϕ(α · a) = φ(α)ϕ(a) (a ∈ A, α ∈ A).

We denote the set of all such maps by ΩA. A bounded linear functional m : A∗ −→ Cis called a module (ϕ, φ)-mean onA∗ if m( f ·a) = φϕ(a)m( f ), m( f ·α) = φ(α)m( f ) andm(φ ϕ) = 1 for each f ∈ A∗, a ∈ A and α ∈ A. We sayA is module (ϕ, φ)-amenableif there exists a module (ϕ, φ)-mean on A∗ [5]. We note that if A = C and φ is theidentity map then the module (ϕ, φ)-amenability coincides with ϕ-amenability [11]. In[5], it is characterized the module (ϕ, φ)-amenability of a Banach algebra A throughvanishing of the first Hochschild module cohomology group H1

A(A, X∗) for certain

∗ speaker

64

H. Ebrahimi,A. Bodaghi

BanachA-bimodules X (for modification of the first Hochschild module cohomologygroup H1

A(A, X∗), by using module homomorphisms between Banach algebras, refer

to [4]).In this paper, we characterize the module (ϕ, φ)-amenability of Banach algebras

through the existence of a bounded net (aγ)γ in A such that ∥aaγ − φ ϕ(a)aγ∥ → 0and ∥α · aγ − φ(α)aγ∥ → 0 for all a ∈ A and α ∈ A. Then, we focus on (ϕ, φ)-meansand establish various criteria for their existence.

2. Second section

These are the main results of the paper.

Proposition 2.1. Let A be a Banach A-bimodule with compatible actions and φ ∈ΦA, ϕ ∈ ΩA. Suppose that for each f ∈ A∗∗ there exists m f ∈ A∗∗ such that∥m f ∥ = ⟨m f , φ ϕ⟩ = 1 and

⟨m f , f · a⟩ = φ ϕ(a)⟨m f , f ⟩, ⟨m f , f · α⟩ = φ(α)⟨m f , f ⟩

for all a ∈ A, α ∈ A. Then,A has a module (ϕ, φ)-mean of norm 1.

Proof. Define

S = m ∈ A∗∗ : ∥m∥ = ⟨m, φ ϕ⟩ = 1 = m ∈ A∗∗ : ∥m∥ ≤ 1; ⟨m, φ ϕ⟩ = 1.

It is easy to check that S is a semigroup with the first Arens product and w∗-compactsubset of A∗∗. Let F denote the collection of all finite subset F ofA∗. For every F ∈ F,we put

S F = m ∈ S : ⟨m, f · a⟩ = φ ϕ(a)⟨m, f ⟩,⟨m, f · α⟩ = φ(α)⟨m, f ⟩, a ∈ A, α ∈ A, f ∈ F.

Then, S F is closed in S and S F1 ⊇ S F2 whenever F1 ⊆ F2. It is obvious that eachm ∈ ∩S F : f ∈ F is a module (ϕ, φ)-mean with ∥m∥ = 1. Now, if we show that S F ,for all F ∈ F, then m ∈ ∩

F∈F S F is the required module mean by finite intersectionproperty. We argue this by induction on number of elements in F. Suppose that somem1 ∈ S F exists and consider g ∈ A∗⧹F. Set h = m1 · g ∈ A∗. By assumption, thereexists m2 ∈ S h such that m = m2m1 ∈ S (since S is a semigroup). For each f ∈ F anda, b ∈ A, we have

⟨m1 · ( f · a), b⟩ = ⟨m1, f · (ab)⟩ = φ ϕ(a)⟨m1, f ⟩φ ϕ(b).

This shows that m1 · ( f · a) = (φ ϕ)(a)⟨m1, f ⟩φ ϕ. Similarly, m1 · f = ⟨m1, f ⟩φ ϕ.So

⟨m, f · a⟩ = ⟨m2m1, f · a⟩ = ⟨m2,m1 · ( f · a)⟩ = φ ϕ(a)⟨m1, f ⟩⟨m2, φ ϕ⟩= φ ϕ(a)⟨m2, ⟨m1, f ⟩φ ϕ⟩ = φ ϕ(a)⟨m2,m1 · f ⟩= (φ ϕ)(a)⟨m2m1, f ⟩ = φ ϕ(a)⟨m, f ⟩,

65

Some Notation of module (ϕ, φ)-amenable Banach algebras

for all f ∈ F and all a ∈ A. Moreover,

⟨m, g · a⟩ = ⟨m2, (m1 · g) · a⟩ = ⟨m2, h · a⟩ = φ ϕ(a)⟨m2, h⟩= φ ϕ(a)⟨m2,m1.g⟩ = φ ϕ(a)⟨m, g⟩

and similarly ⟨m, g · α⟩ = φ(α)⟨m, g⟩ for all a ∈ A, α ∈ A. Also ∥m∥ = ∥m2m1∥ =∥m2∥∥m1∥ = 1. Therefore, m ∈ S F

∪g. This completes the proof.

The upcoming theorem is the main result of the paper which shows that theexistence of module (ϕ, φ)- mean with norm 1 is a pointwise property. In other words,it follows from the existence of an element ofA∗∗ associated with each of the elementsof the ideal ker(φ ϕ).

Recall that a left Banach A-module X is called left essential if the linear span ofA · X = a · x : a ∈ A, x ∈ X is dense in X.

Let A be a Banach A-module with compatible actions and ϕ ∈ Ω(A), φ ∈ ΦA.Then A is module (ϕ, φ)-amenable if and only if there exists a bounded net (aγ) in Asuch that φ ϕ(aγ) = 1 for all γ and

∥aaγ − φ ϕ(a)aγ∥ → 0 and ∥α · aγ − φ(α)aγ∥ → 0

for all a ∈ A and α ∈ A.

Proof. If m is a w∗-cluster point of (aγ), then clearly m satisfies m(φ ϕ) = 1,m( f · a) = φ ϕ(a)m( f ) and m( f · α) = φ(α)m( f ) for all f ∈ A∗, a ∈ A and α ∈ A.

Conversely, let m be a module (ϕ, φ)-mean. Then m is the w∗-limit of some net (bγ)inA with ∥b j∥ → ∥m∥. Then φ ϕ(bγ)→ m(φ ϕ) = 1, and w∗-continuity gives

abγ − φ ϕ(a)bγ → 0 and α · bγ − φ(α)bγ → 0

in the w∗-topology for all a ∈ A and α ∈ A. So the nets (abγ − φ ϕ(a)bγ) and(α · bγ − φ(α)bγ) in A, both converge to 0 weakly for all a ∈ A and α ∈ A. Now takeany finite subsets F = a1, ..., ak and H = α1, ..., αℓ ofA and A, respectively. Let

C = ((aib − φ ϕ(ai)b)ki=1, (α j · b − φ(α j)b)ℓj=1, φ ϕ(b) − 1) : b ∈ A.

Then in the Banach space Ak+ℓ × C, 0 is in the weak closure of C and hence in thenorm closure since C is convex. Thus, given ε > 0, we can find bF,H,ε ∈ A such that∥bF,H,ε∥ ≤ 2∥m∥, say,

∣∣∣φ ϕ(bF,H,ε) − 1∣∣∣ < ε. Moreover, for each a ∈ A and α ∈ A we

have,

∥abF,H,ε − φ ϕ(a)bF,H,ε∥ < ε and ∥α · bF,H,ε − φ(α)bF,H,ε∥ < ε.

Finally replace bF,H,ε by a scalar multiple aF,H,ε = λF,H,εbF,H,ε for which φ ϕ(aF,H,ε) =1. Then

∣∣∣λF,H,ε

∣∣∣ < 11−ε and

∥aaF,H,ε − φ ϕ(a)aF,H,ε∥ <ε

1 − ε and ∥α · aF,H,ε − φ(α)aF,H,ε∥ <ε

1 − ε .

Therefore the net (aF,H,ε) is a bounded approximate module (ϕ, φ)-mean and m is thew∗-limit of (aF,H,ε).

66


Theorem 2.2. LetA be a Banach A-module with compatible actions and φ ∈ ΦA, ϕ ∈ΩA. Consider the following conditions.(i) There exists a module (ϕ, φ)- mean m such that ∥m∥ = 1;(ii) There exists a net (u j) j in A such that φ ϕ(u j) = 1, for all j, ∥u j∥ → 1 and

∥au j∥ →∣∣∣(φ ϕ)(a)

∣∣∣, ∥α · u j∥ →∣∣∣φ(α)

∣∣∣ for all a ∈ A and α ∈ A;(iii) For each a ∈ ker(φ ϕ) and b ∈ ker(ϕ), there exists ma,b ∈ A∗∗ with

∥ma,b∥ ≤ 1, ⟨ma,b, φ ϕ⟩ = 1 and ama,b = bma,b = 0, α · ma,b = φ(α)ma,b forall α ∈ A;

(iv) For each a ∈ ker(φ ϕ), b ∈ ker(ϕ) and ϵ > 0, there exists u ∈ A such that∥u∥ ≤ 1 + ϵ, ∥au∥ ≤ ϵ, ∥bu∥ ≤ ϵ, ∥α · u − φ(α)u∥ ≤ ϵ and φ ϕ(u) = 1 for alla ∈ A, α ∈ A.

Then (iv)⇐(iii)⇐(i)⇒(ii)⇒(iv). If, in addition,A is a left or right essential A-module,then all assertions are equivalent.

Proof. (i)⇒(ii) Let there exists a module (ϕ, φ)-mean m such that ∥m∥ = 1. Then, byProposition 2 there exists a net (u j) j inA with the following properties:

∥u j∥ → ∥m∥ = 1, ∥au j − (φ ϕ)(a)u j∥ → 0, ∥α · u j − φ(α)u j∥ → 0

for all a ∈ A, α ∈ A. Thus∣∣∣∥au j∥ − |(φ ϕ)(a)|∣∣∣ ≤ ∣∣∣∥au j∥ − ∥(φ ϕ)(a)u j∥

∣∣∣+

∣∣∣∥(φ ϕ)(a)u j∥ − ∥(φ ϕ)(a)∥∣∣∣

≤ ∥au j − (φ ϕ)(a)u j∥ + |∣∣∣(φ ϕ)(a)

∣∣∣∥u j∥ − 1∣∣∣

→ 0

and ∣∣∣∥αu j∥ −∣∣∣φ(α)|

∣∣∣ ≤ ∣∣∣∥αu j∥ − ∥φ(α)u j∥∣∣∣ + ∣∣∣∥φ(α)u j∥ −

∣∣∣φ(α)|∣∣∣

≤ ∥αu j − φ(α)u j∥ + |φ(α)|(∥u j∥ − 1)→ 0

Therefore, ∥au j|| →∣∣∣(φ ϕ)(a)

∣∣∣ and ∥αu j|| →∣∣∣φ(α)

∣∣∣ so (ii) holds.(i)⇒(iii) If m is module (ϕ, φ)-mean, we can choose ma,b = m, for all a ∈ker(φ

ϕ), b ∈ker(ϕ), and thus ∥ma,b∥ ≤ 1, ⟨ma,b, φ ϕ⟩ = ⟨m, φ ϕ⟩ = 1. On the other hand,for all f ∈ A∗, we get

⟨ama,b, f ⟩ = ⟨ma,b, f · a⟩ = ⟨m, f · a⟩ = (φ ϕ)(a)⟨m, f ⟩ = 0

and⟨bma,b, f ⟩ = ⟨m, f · b⟩ = φ ϕ(b)⟨ma,b, f ⟩ = φ(0)⟨m, f ⟩ = 0.

Also, ⟨α ·ma,b, f ⟩ = ⟨m, f ·α⟩ = φ(α)⟨ma,b, f ⟩ for all α ∈ A. The above relations implythat ama,b = bma,b = 0 and α · ma,b = φ(α)ma,b for all a ∈ker(φ ϕ), b ∈ ker(ϕ).

(ii)⇒(iv) It is obvious.

67


(iii)⇒(iv) Fix a ∈ker(φ ϕ), b ∈ ker(ϕ) and take any net (u j) j in A such that∥u j∥ ≤ 1 in which u j → ma,b in w∗-topology. Then (φ ϕ)(u j)→ 1. Replacing each u jwith the scaler multiple of itself and taking a coefficient subnet, we may arrange that∥u j∥ ≤ 1 + ϵ and (φ ϕ)(u j) = 1 for all j. We have

w∗ − limj

au j = ama,b = 0, w∗ − limj

bu j = ma,b = 0,

and w∗ − lim j(α · u j − φ(α)u j) = 0 for α ∈ A. Thus, 0 is in the weak closure of sets(au j) j, (bu j) j and (α · u j − φ(α)u j) j. Hence, 0 is in the norm closure of convex hullof the mentioned sets. Thus, the set (u j) j beings contained in the closed hyperplanex ∈ A; (φ ϕ)(x) = 1, we easily arrive our conclusion.

(iv)⇒(i) We claim that for finite subset F,H of A,A and ϵ > 0, there exists uF,H,ϵsuch that (φ ϕ)(uF,H,ϵ) = 1, ∥uF,H,ϵ∥ ≤ 1 + ϵ and for all a ∈ F, α ∈ H

∥a · uF,H,ϵ − (φ ϕ)(a)uF,H,ϵ∥ ≤ ϵ, ∥α · uF,H,ϵ − φ(α)uF,H,ϵ∥ ≤ ϵ

Let F = a1, ..., ak,H = α1, α2, ..., αk, and choose δ > 0 such that (1 + δ)k+1 ≤ 1 + ϵ,by hypothesis , there exists u0 ∈ A such that (φ ϕ)(u0) = 1 and ||u0|| ≤ 1 + δ.SinceA is a left or right essential A-module, it follows from the proof of [6, Theorem3.14] that the map ϕ is C-linear. Thus, α1u0 − φ(α1)u0 ∈ ker(ϕ). On the otherhand, a1u0 − (φ ϕ)(a1)u0 ∈ker(φ ϕ). Again by (iv) there exists u1 ∈ A such that(φ ϕ)(u1) = 1, ∥u1∥ ≤ 1 + δ and

∥(a1u0 − (φ ϕ)(a1)u0)u1∥ ≤ δ, ∥(α1u0 − φ(α1)u0)u1∥ ≤ δ.

Similarly, a2u0u1 − (φϕ)(a2)u0u1 ∈ker(φϕ), α2u0u1−φ(α2)u0u1 ∈ ker(ϕ) and hencethere exists u2 ∈ A such that (φ ϕ)(u2) = 1, ∥u2∥ ≤ 1 + δ and

∥(a2u0u1 − (φ ϕ)(a2)u0u1)u2∥ ≤ δ, ∥(α2u0u1 − φ(α2)u0u1)u2∥ ≤ δ.

Thus for j = 1, 2 we have ∥u j∥ ≤ 1 + δ, (φ ϕ)(u j) = 1 and

∥a ju0u1u2 − (φ ϕ)(a j)u0u1u2∥ ≤ δ(1 + δ), ∥α ju0u1u2 − φ(α j)u0u1u2∥ ≤ δ(1 + δ).

Proceeding inductively, we see there exits u j (1 ≤ j ≤ k) such that (φ ϕ)(u j) =1, ∥u j∥ ≤ 1 + δ and for i = 1, ..., j

∥aiu0u1...u j − (φ ϕ)(ai)u0u1...u j∥ ≤ δ(1 + δ) j−1 ≤ ϵ,

∥αi · u0u1...u j − φ(αi)u0u1...u j∥ ≤ δ(1 + δ) j−1 ≤ ϵ.In particular, when j = k, setting uF,H,ϵ =

∏kj=0 u j gives us (φ ϕ)(uF,H,ϵ) =∏k

j=0(φ ϕ)(u j) = 1 and

∥uF,H,ϵ∥ ≤ ∥u0∥∥u1∥...∥uk∥ ≤ (1 + δ)k+1 ≤ 1 + ϵ.

Also, for each a ∈ F, α ∈ H, we have

∥auF,H,ϵ − (φ ϕ)(a)uF,H,ϵ∥ = ∥au0u1...uk − (φ ϕ)(a)u0u1...uk∥ ≤ δ(1 + δ)k−1 ≤ ϵ,

68


and

∥α · uF,H,ϵ − φ(α)uF,H,ϵ∥ = ∥αu0u1...uk − φ(α)u0u1...uk∥ ≤ δ(1 + δ)k−1 ≤ ϵ.

This proves the above claim. Now, order the Triplet (F,H, ϵ), F ⊆ A,H ⊆ A finiteand ϵ > 0, in the obvious manner manner, and let m be a w∗-cluster point of the net(uF,H,ϵ)F,H,ϵ in A∗∗. Then, ∥m∥ ≤ 1 and ⟨m, φ ϕ⟩ = 1 and thus ∥m∥ = 1 and for alla ∈ A and α ∈ A, we get

⟨m, f · a⟩ = limF,H,ϵ⟨uF,H,ϵ , f · a⟩ = lim

F,H,ϵ⟨a · uF,H,ϵ , f ⟩ = (φ ϕ)(a)⟨m, f ⟩,

and similarly ⟨m, f ·α⟩ = φ(α)⟨m, f ⟩. Therefore, m is required module (ϕ, φ)-mean.

Remark 2.3. Using similar methods to those employed in the proof of above Theorem,the following can be shown: Let A be a Banach A-module with compatible actionsand φ ∈ ΦA ∪ 0, ϕ ∈ ΩA. Then for the following conditions, we have the sameimplications as Theorem 2.2.

(i) A has a module (ϕ, φ)-mean of norm C;(ii) A contain an approximate (ϕ, φ)-mean with norm bounded C;(iii) For each a ∈ker(φ ϕ), b ∈kerϕ, there exists ma,b ∈ A∗∗ with ∥ma,b∥ =

C, ⟨ma,b, φ ϕ⟩ = 1 and ama,b = bma,b = 0and α · ma,b = φ(α)ma,b;(iv) There exists a net (u j) j inA with (φϕ)(u j) = 1, ∥u j∥ → C, for all j and au j → 0,

for every a ∈ker(φ ϕ) and ∥α · u j∥ → |φ(α)| for every α ∈ A.

Let A be a Banach A-module with compatible actions and φ ∈ ΦA ∪ 0, ϕ ∈ ΩA.Consider the set of all f ∈ A∗ with the following property:For each δ > 0, there exists a sequence (an)n inA such that (φϕ)(an) = 1, ∥an∥ ≤ 1+δfor all n, and ∥ f · an∥ → 0. We denote this set by N(A, φ ϕ).

We have the following result which is analogous to Lemmas 2.6 and 2.7 from [12].

Lemma 2.4. Let A be a Banach A-module with compatible actions and φ ∈ ΦA ∪0, ϕ ∈ ΩA. Then, the following hold.

(i) φ ϕ < N(A, φ ϕ).(ii) N(A, φ ϕ) is closed inA∗ and closed under scaler multiplication.(iii) IfA is commutative, then N(A, φ ϕ) is closed under addition.(iv) If A admits a module (ϕ, φ)-mean of norm 1, then N(A, φ ϕ) is subspace of

A∗.

Proof. The proofs of [12, Lemma 2.6] and [12, Lemma 2.7] work verbatim if we putφ ϕ instead of φ in their proofs.

We now aim at a criterion for the existence of module (φ ϕ)-mean of norm 1involving the set N(A, φ ϕ).

Theorem 2.5. Let A be a Banach A-module with compatible actions and φ ∈ ΦA ∪0, ϕ ∈ ΩA. Then the following four condition are equivalent:

69


(i) There exists a module (ϕ, φ)-mean with ∥m∥ = 1;(ii) N(A, φϕ) is subspace ofA∗ and f ·a− f , f ·α− f ∈ N(A, φϕ) for all f ∈ A∗

and all a ∈ A, α ∈ A with (φ ϕ)(a) = 1.

Proof. Let (i) holds. By Lemma 2.4. N(A, φ ϕ) is a subspace of A∗. Let f ∈ A∗and a ∈ A, α ∈ A with (φ ϕ)(a) = 1, φ(α) = 1. By Theorem 2.2 there exists a net(u j) j inA such that (φ ϕ)(u j) = 1, ∥u j|| → 1 and

∥a · u j − (φ ϕ)(a)u j∥ = ∥a · u j − u j|| → 0, ∥α · u j − φ(α)u j∥ = ∥α · u j − u j∥ → 0.

Since ∥( f ·a− f ) ·u j∥ ≤ ∥ f ∥∥au j −u j∥ and ∥( f ·α− f ) ·u j∥ ≤ ∥ f ∥∥α ·u j −u j∥, it followsthat f · a − f , f · α − f ∈ N(A, φ ϕ).

Conversely, suppose that N(A, φ ϕ) is subspace of A∗ and that (ii) holds. Sinceφ ϕ < N(A, φ ϕ) and ||φ ϕ|| = 1, by the Hahn-Banach theorem there existsm ∈ A∗∗ such that ∥m∥ = ⟨m, φ ϕ⟩ = 1 and m|N(A,φϕ) = 0. By assumption, for eacha ∈ A, α ∈ A with (φ ϕ)(a) = 1 and φ(α) = 1, we have

⟨m, f · a⟩ = ⟨a · m, f ⟩ = (φ ϕ)(a)⟨m, f ⟩,

⟨m, f · α⟩ = ⟨α · m, f ⟩ = φ(α)⟨m, f ⟩for all a ∈ A, α ∈ A and f ∈ A∗. This means that m is a module (ϕ, φ)-mean.

Corollary 2.6. Let Banach algebras A and A be Banach algebras. If φ : A −→ Cand ϕ : A −→ A are the classical character and module character, respectively,we showed when A is module (ϕ, φ)-amenable. Moreover, we found the relations ofmodule (ϕ, φ)-amenability ofA and its ideals.

Acknowledgement

Acknowledgment ... The authors sincerely thank the anonymous reviewers for theircareful reading of the manuscript and helpful comments.

[1] M. Amini, Module amenability for semigroup algebras, Semigroup Forum. 69 (2004), 243–254.[2] M. Amini and D. Ebrahimi Bagha, Weak module amenability of semigroup algebras, Semigroup

Forum. 71 (2005), 18–26.[3] A. Bodaghi, Semigroup algebras and their weak module amenability, J. Appl. Func. Anal. 7, No.

4 (2012), 332–338.[4] A. Bodaghi, Module (φ, ψ)-amenability of Banach algebras, Arch. Math. (Brno). 46, No. 4 (2010),

227–235.[5] A. Bodaghi and M. Amini, Module character amenability of Banach algebras, Arch. Math. (Basel).

99 (2012), 353–365.[6] A. Bodaghi, M. Amini and R. Babaee, Module derivations into iterated duals of Banach algebras,

Proc. Romanian. Acad., Series A. 12, No. 4 (2011), 277–284.

70


[7] H. G. Dales, Banach algebras and automatic continuity, London Mathematical Society Mono-graphs 24, Clarendon Press, Oxford, 2000.

[8] F. Gourdeau, Amenablility and the second dual of a Banach algebras, Studia Math. 125 (1) (1997),75–80.

[9] A. T. Lau, Analysis on a class of Banach algebras with applications to harmonic analysis on locallycompact groups and semigroups, Fund. Math. 118 (1983), 161-175.

[10] B. E. Johnson, Cohomology in Banach Algebras, Memoirs Amer. Math. Soc. 127, Providence,1972.

[11] E. Kaniuth, A. T. Lau, and J. Pym, On φ-amenability of Banach algebras, Math. Proc. Camb. Soc.144 (2008), 85–96.

[12] E. Kaniuth, A.T. Lau, J. Pym, On Character Amenability of Banach Algebras, J. Math. Anal. Appl.344 (2008), 942–955.

[13] M. S. Monfared, Character amenability of Banach algebras, Math. Proc. Camb. Soc. 144 (2008),697–706.

[14] H. Pourmahmood and A. Bodaghi, Module approximate amenability of Banach algebras, Bull.Iran. Math. Soc. 39, No. 6 (2013), 1137–1158.

H. Ebrahimi,Department of Mathematics, University of Isfahan, P.O.BOX 81764-73441, Isfahan,Iran, e-mail: [email protected]

A. Bodaghi,Department of Mathematics, Garmsar Branch, Islamic Azad University, Garmsar, Iran,e-mail: [email protected]

71



January 20–21, 2016

OPERATOR ALGEBRAS OF WEIGHTED CONDITIONALEXPECTATION OPERATORS

Y. Estaremi∗

Abstract

In this paper we consider the space of weighted conditional expectation operators on L2(Σ), then wecompute the commutant and double commutants of the algebra of weighted conditional expectationoperators. Finally we prove that a sub-classes of these operators is a Von Neumann algebra.

2010 Mathematics subject classification: Primary 47L80.Keywords and phrases: conditional expectation operator, commutant, commutator, Von Neumannalgebras..

1. Introduction

Let (X,F , µ) be a complete σ-finite measure space. All sets and functions statementsare to be interpreted as holding up to sets of measure zero. For a σ-subalgebra A ofF , the conditional expectation operator associated with A is the mapping f → EA f ,defined for all non-negative f as well as for all f ∈ Lp(F ) = Lp(X,F , µ), 1 ≤ p ≤ ∞,where EA f is the uniqueA-measurable function satisfying∫

A(EA f )dµ =

∫A

f dµ ∀A ∈ A.

We will often write E for EA. This operator will play a major role in our work and welist here some of its useful properties:

• If g isA-measurable, then E( f g) = E( f )g.• If f ≥ 0, then E( f ) ≥ 0; if E(| f |) = 0, then f = 0.• |E( f g)| ≤ (E(| f |p))

1p (E(|g|p′))

1p′ ; p−1 + p′−1 = 1.

• For each f ≥ 0, S (E( f )) is the smallest A-set containing S ( f ), where S ( f ) = x ∈X : f (x) , 0.A detailed discussion and verification of most of these properties may be found in [6].We are concerned here with linear operators acting on Lp(F ) = Lp(X,F , µ) speciallyon L2(F ).∗ speaker

72

Y. Estaremi

We continue our investigation about the class of bounded linear operators on the Lp-spaces having the form MwEMu, where E is the conditional expectation operator, Mw

and Mu are (possibly unbounded) multiplication operators and it is called weightedconditional expectation operator. Our interest in operators of the form MwEMu stemsfrom the fact that such forms tend to appear often in the study of those operators re-lated to conditional expectation. Weighted conditional expectation operators appearedin [1], where it is shown that every contractive projection on certain L1-spaces can bedecomposed into an operator of the form MwEMu and a nilpotent operator.

2. Second section

In this section first we give the definition of weighted conditional expectationoperators on Lp-spaces.

Definition 2.1. Let (X,Σ, µ) be a σ-finite measure space and letA be a σ-subalgebraof Σ such that (X,A, µA) is also σ-finite. Let E be the conditional expectation operatorrelative toA. If 1 ≤ p, q < ∞ and u,w ∈ L0(Σ) (the spaces of Σ-measurable functionson X) such that u f is conditionable and wE(u f ) ∈ Lq(Σ) for all f ∈ D ⊆ Lp(Σ), whereD is a linear subspace, then the corresponding weighted conditional expectation (orbriefly WCE) operator is the linear transformation MwEMu : D → Lq(Σ) defined byf → wE(u f ).

As was proved in [2], the WCE operator MwEMu on Lp(Σ) is bounded if and onlyif (E(|u|p′))

1p′ (E(|w|p))

1p ∈ L∞(A), where 1

p +1p′ = 1. Now we define some notations.

Let 1 < p < ∞ and

Wp =Wp(A) = MwEMu : (E(|u|p′))1p′ (E(|w|p))

1p ∈ L∞(A) u,w ∈ D(E).

Also for w, u ∈ D(E) set

Ww,p =Ww,p(A) = MwEMu : (E(|w|p))1p (E(|u|p′))

1p′ ∈ L∞(A) u ∈ D(E),

Wp,u =Wp,u(A) = MwEMu : (E(|w|p))1p (E(|u|p′))

1p′ ∈ L∞(A) w ∈ D(E).

HenceW1,p = EMu : (E(|u|p′))

1p′ ∈ L∞(A) u ∈ D(E),

Wp,1 = MwE : (E(|w|p′))1p′ ∈ L∞(A) w ∈ D(E).

These observations shows thatWw,p is the adjoint ofWp,w as a subset of Banachspace B(Lp) (the algebra of all bounded operators on Lp), where 1 < p < ∞ andw ∈ D(E). Here we have the conditional-type Minkowski inequality as follows:

73

WEIGHTED CONDITIONAL EXPECTATION OPERATORS

Lemma 2.2. For measurable functions f , g ∈ D(E) and 1 ≤ p < ∞ we have

(E(| f + g|p))1p ≤ (E(| f |p))

1p + (E(|g|p))

1p

Let u, u′,w,w′ ∈ D(E) and α ∈ C. Then we have αw,wE(uw′) ∈ D(E) and thefollowings:• MwEMu Mw′EMu′ = MwE(uw′)EMu′ .• MwEMu + MwEMu′ = MwEMu+u′ .• αMwEMu = MαwEMu = MwEMαu.Now by using conditional-type Holder and conditional-type Minkowski inequalitieswe have the followings hold:• (E(|αw|p))

1p (E(|u|p′))

1p′ = |α|(E(|w|p))

1p (E(|u|p′))

1p′ .

• (E(|w|p))1p (E(|u + u′|p′))

1p′ ≤ (E(|w|p))

1p [(E(|u|p′))

1p′ + (E(|u′|p′))

1p′ ].

• (E(|wE(uw′)|p))1p (E(|u′|p′))

1p′ ≤ (E(|u|p′))

1p′ (E(|w|p))

1p (E(|w′|p))

1p (E(|u′|p′))

1p′ .

Therefore

∥MwEMu Mw′EMu′∥ ≤ ∥MwEMu∥∥Mw′EMu′∥.

and

∥MwEMu + MwEMu′∥ ≤ ∥MwEMu∥ + ∥MwEMu′∥.

These observations imply that:

• Wp is closed with respect to the scalar and operator products.

• The spacesWw,p andWp,u are operator algebra, for all w, u ∈ D(E).

Now, in the next theorem we compute the spectrum of MwEMu and also we give aformula for the inverse of λI − MwEMu.

Theorem 2.3. Let MwEMu ∈ Wp for 1 < p < ∞. Then

σ(MwEMu) \ 0 = ess range(E(uw)) \ 0.

Moreover for each λ ∈ σ(MwEMu) \ 0 we have

(λI − MwEMu)−1 =1λ

I − M( wλ(E(wu)−λ) )EMu.

By Theorem 2.3 we get that the collectionW = λI + T : 0 , λ ∈ C, T ∈ Wp isinvertibly closed. Let

Vp = MwEMu : (E(|w|p), E(|u|p′)) ∈ L∞(A) × L∞(A).

It is clear thatVp ⊆ Wp,W1,p = V1,p andWp,1 = Vp,1, for p > 1.

74

Y. Estaremi

If V is an algebra of bounded operators, then its commutant V′ is the set of allbounded operators that commute with every element inV. In symbols (and in the textof weighted conditional type operators):

(Alg(W2))′ =W′2 = T ∈ B(L2(Σ)) : MwEMuT = T MwEMu,∀(w, u) ∈W2.

In which Alg(W2) is the operator algebra generated by W2. Now we recall that(W1,2)′ = L∞(A), where L∞(A) = Ma : a ∈ L∞(A) [3]. In the next proposition weobtain the commuatnt of Alg(W2).

Proposition 2.4. W′2 = Alg(W2)′ = L∞(A).

By Proposition 2.4 we get that L∞(A)′ = W′′2 and so L∞(A)′ = Alg(W2)′′.

Hence by using theorem 3.2 and corollary 3.3 of [4], we have the next proposition.

Proposition 2.5. Let T be a continuous linear transformation on L2(Σ). Then thefollowings are equivalent:• T ∈ W′′

2 = Alg(W2)′′,• There is a constant C such that, for every f ∈ L2(Σ),

E(|T f |2) ≤ C.E(| f |2) a.e.

• For each f ∈ L2(Σ), there is a constant C f such that

E(|T f |2) ≤ C f .E(| f |2) a.e.

• For each f ∈ L2(Σ), S (T f ) ⊆ S (E(| f |)).• For each f ∈ L2(Σ), define the measure µ f onA by

dµ f = | f |2dµ|A.

Then for all f , µT f ≪ µ f .

If T = MwEMu ∈ W2, then T ∗ = MuEMw. Therefore T ∗ ∈ W2. This implies that(W2)∗ =W2 and (W1,2)∗ =W2,1. Here we recall a fundamental lemma from generaloperator theory [3].

Lemma 2.6. Let T ∈ B(L2(Σ)) and S be a closed operator on L2(Σ). If T = S on adense subset of L2(Σ), then S is bounded and T = S .

Now we recall that: T = EMu ∈ W1,2 is normal if and only if u ∈ L∞(A). LetWN

1,2 = EMu : u ∈ L∞(A).

Proposition 2.7. WN1,2 is a unital commutative Von Neumann algebra with unit E.

Let EMuα be a net in WN1,2. Then easily we get that the net EMuα in WN

1,2 is anapproximate unit if and only if

∥uα − 1∥∞ → 0.

75

WEIGHTED CONDITIONAL EXPECTATION OPERATORS

As is known for every f ∈ L∞(Σ) we have σ( f ) = ess range( f ) and also we have

σ(EMu) \ 0 = ess range(E(u)) \ 0.

Hence if we define the operator Θ as:

Θ : L∞(A)→W1,2, Θ(u) = EMu,

then easily we get that Θ is a unital isometric ∗-homomorphism that preserves thespectrum. Consequently, it is a unital isometric ∗-isomorphism ontoWN

1,2.In the next theorem we prove that the weak∗ convergence of the net uαα in L∞(A) isequivalent to weak operator convergence of the net EMuαα in B(L2(Σ))(The algebraof all bounded linear operators on L2(Σ)).

Theorem 2.8. If (X,Σ, µ) is a σ-finite measure space and uαα is a net in L∞(A), thenuα → 0 weak∗ in L∞(A) if and only if EMuα → 0 WOT in B(L2(Σ)).

Finally we give a representation for the C∗-algebra L∞(A).

Remark 2.9. The map φ : L∞(A) → B(L2(Σ)) defined by φ(u) = EMu is a ∗-homomorphism, i.e., the pair (φ, L2(Σ)) is a representation for L∞(A) as a C∗-algebra.

[1] R. G. Douglas, Contractive projections on an L1 space, Pacific J. Math 15 (1965), 443-462.[2] Y. Estaremi and M.R. Jabbarzadeh, Weighted lambert type operators on Lp-spaces, Oper. Matri-

ces 1 (2013), 101-116.[3] J. Herron, Weighted conditional expectation operators, Oper. Matrices 1 (2011), 107-118.[4] A. Lambert, conditional expectation related characterizations of the commutant of an abelian

W∗-algebra, Far East J. of Math. Sciences 2 (1994), 1-7.[5] E. C. Lance, Hilbert C∗-modules, Cambridge University Press, Cambridge, 1995.[6] M. M. Rao, Conditional measure and applications, Marcel Dekker, New York, 1993.

Y. Estaremi,Department of Mathematics,Payame Noor University,Tehran, Irane-mail: [email protected]

76



January 20–21, 2016

Daws Conjecture on the Arens regularity of B(X)

R. Faal∗ and H. R. Ebrahimi Vishki

AbstractWe focus on a question raised by M. Daws [Bull. London Math. Soc. 36 (2004), 493-503] concerning theArens regularity of B(X), the algebra of operators on a Banach space X. In this respect, we introduce thenotion of ultra-reflexivity for a Banach space and we characterize the Arens regularity of B(X) in termsof the ultra-reflexivity of X.

2010 Mathematics subject classification: 47L10; 47L50; 46H25 .Keywords and phrases: Arens products, algebra of operators, reflexive space, super-reflexive, weakoperator topology, ultrapower.

1. Introduction

The second dual A∗∗ of a Banach algebra A can be made into a Banach algebra withtwo, in general different, (Arens) products, each extending the original product of A[1]. A Banach algebra A is said to be Arens regular when the Arens products coincide.For example, every C∗−algebra is Arens regular [2]. For an explicit description of theproperties of these products and the notion of Arens regularity one may consult with[3].

For the Banach algebra B(X), bounded operators on a Banach space X, Dawsshowed that, if X is super-reflexive then B(X) is Arens regular; [4, Theorem 1]. Healso conjectured the validity of the converse. To the best of our knowledge, it seemsthat this has not be solved yet. It has been, however, known that the Arens regularityof B(X) necessities the reflexivity of X, (for a proof see [3, Theorem 2.6.23]).

We introduce the notion of ultra-reflexive space and compare it with the super-reflexivity. Our main aim is to characterize the Arens regularity of B(X) in terms ofthe ultra-reflexivity of X.

2. Preliminaries

Let X be a Banach space, I be an indexing set and let U be an ultrafilter on I. Wedefine the ultrapower XU of X with respect toU, by the quotient space

XU = ℓ∞(X, I)/NU ,∗ speaker

77

R. Faal and H. R. Ebrahimi Vishki

where ℓ∞(X, I) is the Banach space

ℓ∞(X, I) = (xα)α∈I ⊆ X : ∥(xα)∥ = supα∈I∥xα∥ < ∞,

and NU is the closed subspace

NU = (xα)α∈I ∈ ℓ∞(X, I) : limU∥xα∥ = 0.

Then the norm ∥(xα)∥U := limU ∥xα∥ coincides with the quotient norm. We canidentify X with a closed subspace of XU via the canonical isometric embeddingX → XU , sending x ∈ X to the constant family (x). Ample information aboutultrapowers can be found in [5].

A Banach space X is called super-reflexive if every finitely representable Banachspace in X is reflexive. In the language of ultrapowers, it has been shown that Y isfinitely representable in X if and only if Y is isometrically isomorphic to a subspace ofXU for some ultrafilter U on X; [5, Theorem 6.3]. It follows that a Banach space issuper-reflexive if and only if all of its ultrapowers are reflexive.

As it has been shown in [5, Section 7], there is a canonical isometry J : (X∗)U →(XU)∗ defined by the rule

⟨J(( fα)U), (xα)U⟩ = limU⟨ fα, xα⟩ (( fα)U ∈ (X∗)U , (xα)U ∈ XU),

which is a surjection if and only if XU is reflexive (whereU is countably incomplete).In particular, when X is super-reflexive then J is an isometric isomorphism.

As the Ball(X∗∗) is w∗−compact, one can define a norm-decreasing map σ : XU →X∗∗ by

σ((xα)U) = w∗ − limUκX(xα), ((xα)U ∈ XU),

where κX is the canonical embedding of X into X∗∗. We quote the next result whichestablishes a useful connection between X∗∗ and XU for some ultrafilterU.

Proposition 2.1 ([5, Proposition 6.7]). Let X be a Banach space. Then there exist anultrafilter U and a linear isometric embedding K : X∗∗ → XU such that σ K is theidentity on X∗∗ and K κX is the canonical embedding of X into XU . Thus K σ is anorm-1 projection of XU onto K(X∗∗).

3. The main result

We recall that the super-reflexivity of X is equivalent to that of ℓ2(X), the Banachspace of all 2−summable sequences in X, (see [4, Proposition 4]). So X is super-reflexive if and only if Ball(ℓ2(X)U) is weakly compact. This motivates to introducethe notion of ultra-reflexivity in the next definition.

Definition 3.1. A Banach space X is called ultra-reflexive if Ball(B(X))(xU) is weaklycompact for every ultra-filterU and each xU ∈ ℓ2(X)U .

78

Daws Conjecture on the Arens regularity of B(X)

It is obvious that every ultra-reflexive space X is reflexive. Indeed, for each non-zero x ∈ X, B(X)(x) = X. It is also worth to note that, if X is super-reflexive thenX is ultra-reflexive. Therefore ultra-reflexivity lies between reflexivity and super-reflexivity.

In the following we present our main result characterizing the Arens regularity ofB(X) in terms of the ultra-reflexivity of X.

Theorem 3.2. For a Banach space X the following assertions are equivalent.(a) B(X) is Arens regular.(b) fU Ball(B(X)) is w∗−compact for every ultrafilterU and each fU ∈ ℓ2(X)U

∗.

(c) fU Ball(B(X)) is w−compact for every ultrafilterU and each fU ∈ ℓ2(X)U∗.

(d) X is ultra-reflexive.

Since for every super-reflexive space X the algebra B(X) is Arens regular (see [4,Theorem 1]), it concluds that every super-reflexive space is ultra-reflexive. However,to the best of our knowledge, we do not know an ultra-reflexive space which is notsuper-reflexive! An example of a reflexive space which is not ultra-reflexive has beenpresented by Daws in [4, Corollary 2].

[1] R. Arens, The adjoint of a bilinear operation, Proc. Amer. Math. Soc. 2 (1951), 839-848.[2] P. Civin and B. Yood, The second conjugate space of a Banach algebra as an algebra, Pacific J.

Math. 11 (1961), 847-870.[3] H.G. Dales, Banach algebras and automatic countinuity, Clarendon Press, Oxford, (2000).[4] M. Daws, Arens regularity of the algebra of operators on a Banach space, Bull. London Math.

Soc. 36 (2004), 493-503.[5] S. Heinrich, Ultraproducts in Banach space theory, J. Reine Angew. Math. 313 (1980), 72-104.

R. Faal,Department of Pure Mathematics,Ferdowsi University of Mashhad, P.O. Box 1159,Mashhad 91775, IRAN.e-mail: [email protected]

H. R. Ebrahimi Vishki,Department of Pure Mathematics and Centre of Excellence in Analysis on AlgebraicStructures (CEAAS),Ferdowsi University of Mashhad, P.O. Box 1159,Mashhad 91775, IRAN.e-mail: [email protected]

79



January 20–21, 2016

Projectivity of c0(S, ω−1)

E. Feizi and J. Soleymani∗

Abstract

Let S be a weakly cancellative inverse semigroup and ω be a weight function on S with condition ω ≥ 1.In this paper we show that c0(S , ω−1) is projective as a ℓ1(S , ω)-module if and only if S is finite.

2010 Mathematics subject classification: 20M50,18G05,46H25.Keywords and phrases: semigroup, Beurling algebra, projective module.

1. Introduction and Preliminaries

Homology properties of some famous class of Banach algebras such as L1(G), C0(G)and L∞(G) has been studied and expounded by H. G. Dales and M. E. Polyakov[1]undertook a study of these properties for locally compact group G. In this paper weinvestigate relationship between the projectivity of c0(S , ω−1) and semigroup S relatedto the work of Ramsden [3]. In fact for a weakly cancellative inverse semigroup S anda weight ω on it with condition ω ≥ 1 we show that c0(S , ω−1) is projective in ℓ1(S , ω)-mod if and only if it is finite. Before, we introduce some definitions and notations.

Let S be a semigroup, we define the sets [st−1] := u ∈ S : ut = s and[t−1s] := u ∈ S : tu = s, where s, t ∈ S . An element t ∈ S is left cancellable if|[t−1s]| ≤ 1 for s ∈ S . Right cancellable elements are defined similarly. The semigroupS is cancellative if each element is both left and right cancellative. The semigroup S isweakly left (respectively, right) cancellative if [t−1s] (respectively, [st−1]) is finite foreach s, t ∈ S and weakly cancellative if it is both weakly left cancellative and weaklyright cancellative. We denote the forced unitization of S by S ♯.

Let S be a semigroup, and let s ∈ S . An element s∗ ∈ S is an inverse of s ifss∗s = s and s∗ss∗ = s∗. In general an inverse will not be unique. An element s ∈ Sis regular if there exists t ∈ S with sts = s. The semigroup S is called regular if everyelement of S is regular.

Definition 1.1. Let S be a semigroup. Then S is called an inverse semigroup if S isregular and every element has a unique inverse.∗ speaker

80

E. Feizi and J. Soleymani

A weight on a semigroup S is a function ω : S → (0,∞) such that ω(st) ≤ω(s)ω(t), for all s, t ∈ S .

Let ω be a weight on S , then the space

ℓ1(S , ω) = f : S → C| f =∑s∈S

f (s)δs, || f ||1,ω =∑s∈S| f (s)|ω(s) < ∞

(where δs ∈ ℓ1(S , ω) is the point mass at s) equipped with the multiplication:

f ⋆ g =∑s∈S

(∑st=r

f (s)g(t)δr)

(and also define f ⋆ g = 0, if for each r ∈ S the equation st = r has no solution;) is aBanach algebra which will be called a weighted semigroup algebra. The dual space ofℓ1(S , ω) is the space

ℓ∞(S , ω−1) = ϕ : S → C|∑s∈S

ϕ(s)es||ϕ||∞,ω = sups∈S| ϕ(s)ω(s)| < ∞

(where es ∈ ℓ∞(S , ω−1) is the point mass at s) whose duality is given by

⟨ϕ, f ⟩ =∑s∈S

ϕ(s) f (s), ϕ ∈ ℓ∞(S , ω−1), f ∈ ℓ1(S , ω).

In particular, ⟨es, δt⟩ = δs,t, where δs,t is the Kronecker delta, that is,

δs,t =

1 s = t

0 s , t

For subset F of S we write χF for the characteristic function of F.

Lemma 1.2. Let S be a semigroup, then es · δt = χ[t−1 s] and δt · es = χ[st−1] , for everys, t ∈ S .

By the latter lemma, for f ∈ ℓ1(S , ω) and ϕ ∈ ℓ∞(S , ω−1), with f =∑

s∈S f (s)δs

and ϕ =∑

s∈S ϕ(s)es, we have:

ϕ · f =∑s,t∈S

ϕ(s) f (s)χ[t−1 s] and f · ϕ =∑s,t∈S

ϕ(s) f (s)χ[st−1]

Definition 1.3. For semigroup S and weight ω on S we define:

c0(S , ω−1) = ϕ : ϕ ∈ ℓ∞(S , ω−1) such thatϕ

ω∈ c0(S )

Then c0(S , ω−1) is a closed subspace of ℓ∞(S , ω−1); indeed, c0(S , ω−1) is the closedlinear span of es : s ∈ S in ℓ∞(S , ω−1). The dual space of c0(S , ω−1) is ℓ1(S , ω).

Note: δt · f (s) = f (st).

81

Projectivity of c0(S , ω−1)

Proposition 1.4. Let S be a semigroup. Then c0(S , ω−1) is a left [right] ℓ1(S , ω)-submodule of ℓ∞(S , ω−1) if and only if S is weakly right [left ] cancellative.

Definition 1.5. Let (X, ||.||X) be a Banach space and ω a weight on S then we defineℓ1ω(S , X) as:

ℓ1ω(S , X) = f : S → X| f =

∑s∈S

f (s)δs, || f ||X1,ω =∑s∈S|| f (s)||ω(s) < ∞

with convolution product as defined in [2] this space is Banach space.

Proposition 1.6. Let S be a semigroup, let ω be a weight on S , let A be a Banachalgebra, and let X be a Banach leftA-module then:

(i) ℓ1ω(S ,A) becomes a Banach algebra with the convolution multiplication.

(ii) ℓ1ω(S , X) becomes a Banach left ℓ1

ω(S ,A)-module with the convolution product.

Let (X, ||.||X) be a Banach space, let ω be a weight on S , and let ℓ1ω(S )⊗X be the

projective tensor product of ℓ1ω(S ) and X.

Lemma 1.7. There is a map αX from ℓ1ω(S , X) into ℓ1

ω(S )⊗X which is isomorphicisomorphism.

Proof. Let f ∈ ℓ1ω(S ) and x ∈ X, and consider the function fx : S → X defined by

fx(s) = f (s)x for s ∈ S and x ∈ X, clearly fx ∈ ℓ1ω(S , X) and || fx|| = || f |||x||. Hence the

map αX : ℓ1ω(S )×X → ℓ1

ω(S , X) defined by αX( f ⊗ x) = fx is well defined, bilinear and

||αX( f ⊗ x)|| = || f ||||x||.

Therefore by continuity there is a unique operator denoted also by αX from ℓ1ω(S )⊗X

into ℓ1ω(S , X) such that αX( f ⊗ x)(s) = f (s)x, αX is isometric linear isomorphism.

Suppose that A is a Banach algebra and that X is a Banach left A-module. Thenℓ1ω(S )⊗A turns into a Banach algebra along with the action specified by:

( f ⊗ a)(g ⊗ b) = f ⋆ g ⊗ ab, f , g ∈ ℓ1ω(S ) and a, b ∈ A.

and ℓ1ω(S )⊗X turns into a Banach left ℓ1

ω(S )⊗A-module with the action specified by:

( f ⊗ a)(g ⊗ x) = f ⋆ g ⊗ a · x, f , g ∈ ℓ1ω(S ), a ∈ A and x ∈ X.

Theorem 1.8. Let S be a semigroup, let ω be a weight on S , let A be a Banachalgebra, and let X be a BanachA-module. Then:

(i) For every u ∈ ℓ1ω(S )⊗A and v ∈ ℓ1

ω(S )⊗X:

αX(uv) = αA(u) ⋆ αX(v).

(ii) αA is an isometric algebraic isomorphism from ℓ1ω(S )⊗A onto ℓ1

ω(S ,A).

82


Proposition 1.9. Let S be a semigroup, then ℓ1ω(S )♯ ℓ1

ω(S ♯)

Proof. It can easily to seen that the map

ψ( f , α)(s) :=

α s = e

f (s) s , e

where e is unit of S ♯ is isometric isomorphism.

2. Projectivity

Definition 2.1. Let A be a Banach algebra a left (right) A-module X is calledprojective if the product map πX : A♯⊗X → X defined by π(a ⊗ x) = a · x, (a ∈A♯, x ∈ X) has right inverse A-module homomorphism whetre A♯ is unit linked ofBanach algebraA.

Lemma 2.2. Let S be weakly right cancellative semigroup such that c0(S , ω−1) is pro-jective in ℓ1(S , ω)-mod with splitting morphism ρ : c0(S , ω−1)→ c0(S , ω−1)⊗ℓ1(S ♯, ω)then there exists an element x ∈ c0(S , ω−1) such that 1 ≤ ||ρ(x)||π.

Proof. Choose t, r ∈ S with [tr−1] , ∅ since this set is finite so there is a s ∈ [tr−1]such that ω(s) ≤ ω(v) for all v ∈ [tr−1]. Now define x = δr · etω(s) then1 = ||x||ω∞ = ||π ρ(x)||ω∞ ≤ ||ρ(x)||π.

Lemma 2.3. Let S be an infinite, weakly right cancellative semigroup such that,for every finite set F ⊂ S , there exists r ∈ S with rS ♯ ∩ F = ∅. Suppose thatc0(S , ω−1) is projective in ℓ1(S , ω)-mod with splitting morphism ρ : c0(S , ω−1) →c0(S , ω−1)⊗ℓ1(S ♯, ω). Then for each N ∈ N, there exist elements x1, · · · , xN inc0(S , ω−1) and a partition F1, · · · , FN of S with the properties:

(i) ||∑Ni=1 xi||ω∞ = 1

(ii) 1 ≤ ||ρ(xi)||π for 1 ≤ i ≤ N

(iii) ||χFiρ(xi) − ρ(xi)||π ≤ 1/3i for 1 ≤ i ≤ N.

Proof. By above lemma we can find a x1 ∈ c0(S , ω−1), such that 1 ≤ ||ρ(x1)||π in theother hand by theorem 1.8 we have c0(S , ω−1)⊗ℓ1(S ♯, ω) ℓ1

ω(r1S ♯, c0(S , ω−1)) so wecan take F1 ∈ r1S ♯ with ||χF1ρ(x1) − ρ(x1)||π ≤ 1/3 by repeating of this argumentsimilarly to the rest of proof [3, Lemma 6.1] the result follows.

Theorem 2.4. Let S be an infinite, weakly right cancellative semigroup. Suppose thatc0(S , ω−1) is projective in ℓ1(S , ω)-mod. Then there exists a finite set F ⊂ S such that,for each r ∈ S , rS ♯ ∩ F , ∅.

Proof. Since c0(S , ω−1) is projective so there is a splitting morphism ρ from c0(S , ω−1)into c0(Sω−1)⊗ℓ1(S ♯, ω) so by lemma 2.3 and similarly to the proof of [3, Theorem6.2] result follows.

83

Projectivity of c0(S , ω−1)

Lemma 2.5. Let S be an infinite, weakly right cancellative semigroup. Suppose thatc0(S , ω−1) is projective in ℓ1(S , ω)-mod. Then there exists an element t ∈ S such that,for every finite set F ⊂ S , there exists r ∈ S \ F with [tr−1] , ∅.Theorem 2.6. Let S be a weakly cancellative inverse semigroup. Then c0(S , ω−1) isprojective in ℓ1(S , ω)-mod if and only if S is finite.

[1] H. G. Dales andM. E. Polyakov, Homological properties of modules over group algebras, Proc.London Math. Soc, 89 (2004) 390-426.

[2] E. Samei, Weak amenability and 2-weak amenability of Beurling algebras, J. Math. Anal. Appl.346 (2008), 451467.

[3] P. Ramsden, Homological properties of modules over semigroup algebras, Journal of FunctionalAnalysis, 258 (2010) 39884009.

E. Feizi,Department of Mathematics,University of Bu-Ali Sina,Hamedan, Iran.e-mail: [email protected]

J. Soleymani,Department of Mathematics,University of Bu-Ali Sina,Hamedan, Iran.e-mail: [email protected]

84



January 20–21, 2016

Homological properties of some class of Fréchet algebras

E. Feizi∗ and J. Soleymani

AbstractLet G be a locally compact group and ωn be a sequence of decreasing continuous weight functions onG with condition ωn ≥ 1 for all n ∈ N. We introduce the Fréchet algebras L∞(ω) = ∩n∈NL∞(G, 1

ωn),

L∞0 (ω) = ∩n∈NL∞0 (G, 1ωn

) and C0(ω) = ∩n∈NC0(G, 1ωn

). The aim of this talk is to study the homologyproperties of these Fréchet algebras related to G. For example, we show that the Fréchet algebra L∞(ω) isprojective if and only if every Banach algebra L∞(G, 1

ωn) is projective for all n ∈ N. Similarly these will

be shown for Fréchet algebras L∞0 (ω) and C0(ω).

2010 Mathematics subject classification: Primary 13D07, 43A20 , Secondary 43A15, 46A04, 46A13.Keywords and phrases: Fréchet algebra, Beurling algebra, projective module, weight function, projectivelimit.

1. IntroductionHomology property of algebras first introduced by Taylor[11] on topology algebras,then after him it developed by Helemskii[6]. Injective, projective and flat are asuch properties that have direct relationship to important properties contractible andamenable spaces. Before introducing the homology property of Fréchet algebras inpaper we investigate some elementary concept of Banach algebras module.

A complete Hausdorff locally convex space A is called topological algebra[6] ifits product is jointly continuous and a complete Hausdorff locally convex space X iscalled a left A-module if it is an algebraic left module over A and if in addition theaction m : A × X → X is jointly continuous. A complete topological algebraA whosetopology is given by an increasing countable family of sub-multiplicative semi-normsis called a Fréchet algebra. Let A be a Banach algebra and X a Fréchet algebra thenwe denote B(A, X) for all continuous morphism with respect to the strong topology,that is,B(A, X) = T : A → X : Pn(T x) ≤ Cn||x|| for all n ∈ N, x ∈ A and some Cn ∈ R

where Pn is family of sub-multiplicative semi-norms of X. This space is a Fréchetalgebra with semi-norm:

Qn(T ) = sup||x||≤1|Pn(T x)|, n ∈ N.

∗ speaker

85


Let G be a locally compact group and ω a weight on it that is a positive continuousfunction with ω(xy) ≤ ω(x)ω(y) for all x, y ∈ G and ω(eG) = 1 where eG is the identityof group G, then the spaces L∞(G, 1

ω) and L1(G, ω) will be defined by:

L∞(G,1ω

) = f Borel measurable : ess supx∈G

| f (x)|ω(x)

< ∞

andL1(G, ω) = f Borel measurable :

∫G| f (x)|ω(x)dm(x) < ∞

By additional hypothesis that f and g in L∞(G, 1ω

) are equals if they are equals locallyalmost every where respect to the left Haar measure m on G. Similarly they are equalsin L1(G, ω) if they are equals almost every where.L∞(G, 1

ω) and L1(G, ω) respectively

with norms:|| f ||∞,ω = ess sup

x∈G

| f (x)|ω(x)

and|| f ||ω =

∫G| f (x)|ω(x)dm(x)

are Banach spaces. L1(G, ω) with convolution product that is f ∗g(x) =∫

G f (y)g(y−1x)dm(y)is a topological algebra and L∞(G, 1

ω) with product f · g = g ∗ f where f (x) = f (x−1)

for all x ∈ G is a left Banach L1(G, ω)-module. L∞(G, 1ω

) is dual space of L1(G, ω)defined by:

⟨ f , g⟩ =∫

Gf (x)g(x)dm(x) f ∈ L1(G, ω), g ∈ L∞(G,

1ω

)

In the other hand similarly to [3, Proposition 7.17] when ω(x) ≥ 1 for all x ∈ G thenL1(G, ω) · L∞(G, 1

ω) = LUC(G, 1

ω) where:

LUC(G,1ω

) = f ∈ L∞(G,1ω

) :fω∈ LUC(G)

and RUC(G) is the set of left uniformly continuous function. Similarly C0(G, 1ω

) willbe defined by:

C0(G,1ω

) = f ∈ L∞(G,1ω

) :fω∈ C0(G)

where C0(G) is the space of all continuous functions f that vanish out of a compactset that is for ϵ > 0 there is a compact subset K of G for which |g(x)| < ϵ for all x inthe complement of K. C0(G, 1

ω) is a closed subspace of L∞(G, 1

ω) that is a left Banach

L1(G, ω)-module.Similarly to the [1] we define L∞0 (G, 1

ω) as closed subspace of L∞(G, 1

ω) consisting

of all g that vanish out of a compact set.

Lemma 1.1. Let f ∈ L1(G, ω) and g ∈ L∞0 (G, 1ω

) then f · g ∈ L∞0 (G, 1ω

) furthermore ifω(x) ≥ 1 for all x ∈ G then f · g ∈ C0(G, 1

ω).

86

Homological properties

Proof. Since | fω| ∈ L1(G) and | gω| ∈ L∞0 (G) clearly | g

ω| ∗ | fω| ∈ L∞0 (G). Now because

1ω(x) ≤

ω(xy−1)ω(y−1) , so:

|g ∗ f (x)ω(x)

| = |∫

G

g(y) f (y−1x)ω(x)

dm(y)| (1)

= |∫

G

g(y)) f (xy−1)ω(x)

dm(y)| (2)

≤∫

G| g(y)ω(y)

f (x−1y)ω(x−1y)|dm(y) = | gω| ∗ | fω|(x) (3)

and therefore g ∗ f ∈ L∞0 (G, 1ω

). Furthermore when f ∈ L1(G, ω), g ∈ L∞0 (G, 1ω

) andω(x) ≥ 1 for all x ∈ G then g ∗ f ∈ LUC(G, 1

ω) so in this situation f ·g ∈ C0(G, 1

ω).

Now suppose that Eαα∈Λ be a family of topological algebras over a directed setΛ and fαβα,β∈Λ a family of morphism from Eβ into Eα with condition:

fαα = idEα, fαγ = fαβ fβγ

for any α, β, γ in Λ, with α ≤ β ≤ γ. Where idEα, is the identity map on Eα.

Then (Eα, fαβ) is called a projective system of topological algebras. With respectto this system the closed subalgebra E of Πα∈ΛEα will be defined by:

E = x = (xα) ∈ F : xα = fαβ(xβ), i fα ≤ β

That is called algebra projective limit of projective system of (Eα, fαβ) that is denotedby E = lim

←−(Eα, fαβ). If for all α ∈ Λ, Eα has identity 1α then E will have identity

(1α)α∈Λ and if fα be the restriction map of projection map πα on E then fα = fαβ fβfor all α ≤ β that induce projective topology on E (see [8]).

In the special case when Enn∈N is a sequence of Banach algebras with conditionEn ⊇ En+1 for all n ∈ N then with fnm : Em → En, (n ≤ m) as inclusion maps (En, fnm)is projective system of Banach algebras En so by [8, Theorem 3.1] E = lim

←−(En, fnm) is

a Fréchet algebra and it is easily seen that E is isomorphic to ∩n∈NEn.

Lemma 1.2. Let (Eα, fαβ) be projective system of topological algebra that for α ∈ Λ,Eα is topological A-module then by pointwise product E = lim

←−(Eα, fαβ) is topological

A-module.

Proof. Since for all β ∈ Λ, a ∈ A and x = (xα) ∈ E we have fβ(a · x) = fβ((a · xα)) =a · fβ(x) = a · xβ by [10, Theorem 5.2] module product is continuous.

As corollary ∩n∈NEn is left FréchetA-module for left BanachA-module Eα

A left A-module X is called faithful if for x ∈ X and a · x = 0 for all a ∈ A we havex = 0.

Lemma 1.3. If in the projective system (Eα, fαβ) all Eα be a left topological A-moduleand faithful then E = lim

←−(Eα, fαβ) is faithful.

87


Proof. Since for β ∈ Λ, a ∈ A and x = (xα) ∈ E we have fβ(a · x) = a · fβ(x) = a · xβso if a · x = 0 then a · xβ = 0 because Eβ is faithful so xβ = 0 for all β ∈ Λ and hencex = 0.

Note. In the rest of this paper we suppose that ωn is decreasing sequnce of weightfunction with condition ωn(x) ≥ 1, for all x ∈ G, and n ∈ N.

2. Main Results (Projectivity)

Definition 2.1. Let A be a Banach algebra a left (right) A-module X is calledprojective if the product map πX : A♯⊗X → X defined by π(a ⊗ x) = a · x, (a ∈A♯, x ∈ X) has right inverse A-module homomorphism whetre A♯ is unit linked ofBanach algebraA.

Theorem 2.2. Let A be a Banach algebra and X be projective left Fréchet A-moduleand at least one of the spaceA, X have approximation property then for any 0 , x ∈ Xthere exists an leftA-module morphism ψ : X → A♯ such that ψ(x) , 0

Proof. Let ρ : X → A♯⊗X such that πXρ = idX and κ : A♯ → A canonical projectionand consider the diagram

X[r]ρA♯⊗X[d]κ⊗idX [r]idA♯⊗ fA♯⊗C[d]κ⊗idA⊗X[r]idA♯⊗ f [ul]πXA♯⊗C

where f is some functional on X.Consider u = (κ ⊗ idX)ρ(x) ∈ A⊗X since x , 0 then πX ρ(x) , 0 so ρ(x) , 0

and therefore u , 0 because of approximation property A or X by Proposition 3.2of [5] there is a f ∈ X′b with strong topology such that (idA ⊗ f )u , 0 let ψ =(idA♯⊗ f )ρ : X → A♯⊗C since the right hand side of the above diagram is commutativeso 0 , (idA ⊗ f )u = (idA ⊗ f ) (κ ⊗ idX)ρ(x) ∈ A⊗X = (κ ⊗ id) (idA♯ ⊗ f )ρ(x) andtherefore (idA♯ ⊗ f )ρ(x) , 0 so ψ(x) , 0 it is easy to see that ψ is left A module andsinceA♯⊗C A♯ the result follows.

Lemma 2.3. Let G be locally compact group and E be projective Fréchet L1(G, ω1)-module which satisfy C00(G) ⊆ E then G is compact.

Proof. It follows by applying the above lemma on the second part of the Theorem 3.1of [4].

Corollary 2.4. Let G be a locally compact group, then one of ∩n∈NC0(G, 1ωn

) or∩n∈NL∞0 (G, 1

ωn) or ∩n∈NL∞(G, 1

ωn) is left L1(G, ω1) projective if and only if G is

compact.

Theorem 2.5. Let G be a locally compact group, then ∩n∈NL∞(G, 1ωn

) is projective ifand only if G is finite.

88

Homological properties

Proof. Let G be finite then for all n ∈ N, L∞(G, 1ωn

) = L∞(G) so ∩n∈NL∞(G, 1ωn

) isprojective by Theorem 3.3 of [4].

Suppose ∩n∈NL∞(G, 1ωn

) be projective then by lemma 2.3 G is compact, so forall n ∈ N, L∞(G, 1

ωn) = L∞(G) and therefore ∩n∈NL∞(G, 1

ωn) = L∞(G) so L∞(G) is

projective and by Theorem 3.3 of [4] G is finite.

Proposition 2.6. Let G be a locally compact group and ω(x) ≥ 1 for all x ∈ G thenL1(G, ω) is projective right L1(G) module.

Proof. By proposition 3.1 of [7] L1(G, ω)⊗L1(G, ω) is isometric isomorphic to L1(G×G, ω × ω) and the rest of proof is similar to the proof of Theorem 3.3.32 of [2].

[1] F. Bahrami, R. Nasr-Isfahani and S. Soltani Renani, Homological properties of certain Banachmodule over group algebras, Proceeding of the Edinburgh Math. Soc, 54 (2011) 321-328.

[2] H.G. Dales, Banach algebras and automatic continuity, Clarendon Press, Oxford, 2000.[3] H. G. Dales and A. T-M. Lau, The second duals of Beurling algebras, Memoirs Amer. Math. Soc,

117 (2005).[4] H. G. Dales andM. E. Polyakov, Homological properties of modules over group algebras, Proc.

London Math. Soc, 89 (2004) 390-426.[5] E. Feizi and J. Soleymani, Contractibility of ultrapower of Fréchet algebras, Homology, Homotopy

and Applications, submitted.[6] A.Ya. Helemskii, The homology of Banach and topological algebras, Moscow University Press,

(1986).[7] A. Mahmoodi, Some properties for Beurling algebras, Methods of Functional Analysis and

Topology, 15 (2009) 259-263.[8] A. Mallios, Topological algebras. Selected topics, North-Holland Mathematics Studies 124,

Amsterdam, (1986).[9] V. Runde, Lectures on amenability, 1774, Springer-Verlag, New York, 2002.

[10] H. Schaefer, Topological vector spaces, The Macmillan Co, New York, 1966.[11] J. L. Taylor, Homology and cohomology for topological algebras, Adv. Math, 9 (1972) 137-182.

E. Feizi,Department of Mathematics,University of Bu-Ali Sina,Hamedan, Iran.e-mail: [email protected]

J. Soleymani,Department of Mathematics,University of Bu-Ali Sina,Hamedan, Iran.e-mail: [email protected]

89



January 20–21, 2016

THE AMINABILITY OF BANACH ALGEBRAS

K. Haghnejad Azar

Abstract

In this paper, by using the Arens regularity of module actions, for Banach algebra A, we establish somerelationships between cohomology groups A and A(2n) with some applications in the n−weak amenabilityof Banach algebras that introduced by Dales, Ghahramani and Grønbæk. For a Banach A − bimodule Band n ≥ 0, if the topological center of the left module action πℓ : A × B→ B of A(2n) on B(2n) is B(2n) andH1(A(2n+2), B(2n+2)) = 0, then H1(A, B(2n)) = 0. We establish the relationships between cohomologicalgroups H1(A, B(n+2)) and H1(A, B(n)), spacial H1(A, B∗) and H1(A, B(2n+1)).

Keywords and phrases: Amenability, weak amenability, n-weak amenability, cohomology groups,derivation, Arens regularity.

1. Introduction

Let B be a Banach A − bimodule. A derivation from A into B is a bounded linearmapping D : A→ B such that

D(xy) = xD(y) + D(x)y f or all x, y ∈ A.

The space of continuous derivations from A into B is denoted by Z1(A, B). Easyexample of derivations are the inner derivations, which are given for each b ∈ B by

δb(a) = ab − ba f or all a ∈ A.

The space of inner derivations from A into B is denoted by N1(A, B). For BanachA − bimodule, B, the quotient space H1(A, B) = Z1(A, B)/N1(A, B) is called the firstcohomology group of A with coefficients in B. The Banach algebra A is said to bea amenable, when for every Banach A − bimodule B, the inner derivations are onlyderivations existing from A into B∗. It is clear that A is amenable if and only ifH1(A, B∗) = Z1(A, B∗)/N1(A, B∗) = 0. A Banach algebra A is said to be a weaklyamenable, if every derivation from A into A∗ is inner. As is well-known [1], the seconddual A∗∗ of Banach algebra A endowed with the either Arens multiplications is aBanach algebra. The constructions of the two Arens multiplications in A∗∗ lead usto definition of topological centers for A∗∗ with respect to both Arens multiplications.In this paper, we extended some propositions from [6, 7, 10] into general situations.

90

K. Haghnejad Azar

Let B be a Banach A − bimodule and let n ≥ 1. Suppose that B(n) is an n − th dualof B. Then B(n) is also Banach A − bimodule, that is, for every a ∈ A, b(n) ∈ B(n) andb(n−1) ∈ B(n−1), we define

⟨b(n)a, b(n−1)⟩ = ⟨b(n), ab(n−1)⟩,

and⟨ab(n), b(n−1)⟩ = ⟨b(n), b(n−1)a⟩.

2. Cohomological groups of Banach algebras

Let B be a Banach A − bimodule and let A(n) and B(n) be n − th dual of A and B,respectively. By [27, page 4132-4134], if n ≥ 0 is an even number, then B(n) is aBanach A(n) − bimodule. Then for n ≥ 2, we define B(n)B(n−1) as a subspace of A(n−1),that is, for all b(n) ∈ B(n), b(n−1) ∈ B(n−1) and a(n−2) ∈ A(n−2) we define

⟨b(n)b(n−1), a(n−2)⟩ = ⟨b(n), b(n−1)a(n−2)⟩.

If n is odd number, then for n ≥ 1, we define B(n)B(n−1) as a subspace of A(n), that is,for all b(n) ∈ B(n), b(n−1) ∈ B(n−1) and a(n−1) ∈ A(n−1) we define

⟨b(n)b(n−1), a(n−1)⟩ = ⟨b(n), b(n−1)a(n−1)⟩.

and if n = 0, we take A(0) = A and B(0) = B.We also define the topological centers of module actions of A(n) on B(n) as follows

ZℓA(n) (B(n)) = b(n) ∈ B(n) : the map a(n) → b(n)a(n) : A(n) → B(n)

is weak∗ − to − weak∗ continuous

ZℓB(n) (A(n)) = a(n) ∈ A(n) : the map b(n) → a(n)b(n) : B(n) → B(n)

is weak∗ − to − weak∗ continuous.

Theorem 2.1. Let B be a Banach A − bimodule and suppose that every derivationD : A∗∗ → B∗ is weak∗ − to−weak∗ continuous. If Zℓ

B∗∗(A∗∗) = A∗∗ and H1(A, B∗) = 0,

then H1(A∗∗, B∗) = 0.

Theorem 2.2. Let B be a Banach A − bimodule and D : A → B(2n) be a continuousderivation. Assume that Zℓ

A(2n) (B(2n)) = B(2n). Then there is a continuous derivationD : A(2n) → B(2n) such that D(a) = D(a) for every a ∈ A.

Corollary 2.3. Let B be a Banach A − bimodule and n ≥ 0. If ZℓA(2n) (B(2n)) = B(2n) and

H1(A(2n+2), B(2n+2)) = 0, then H1(A, B(2n)) = 0.

Corollary 2.4. [6]. Let A be a Banach algebra such that A(2n) is Arens regular andH1(A(2n+2)), A(2n+2)) = 0 for each n ≥ 0. Then A is 2n − weakly amenable.

91

Amenability

Assume that A is Banach algebra and n ≥ 0. We define A[n] as a subset of A asfollows

A[n] = a1a2...an : a1, a2, ...an ∈ A.We write An the linear span of A[n] as a subalgebra of A.

Theorem 2.5. Let A be a Banach algebra and n ≥ 0. Let A[2n] dense in A and supposethat B is a Banach A − bimodule. Assume that AB∗∗ and B∗∗A are subsets of B. IfH1(A, B∗) = 0, then H1(A, B(2n+1)) = 0.

Corollary 2.6. Let A be a Banach algebra with bounded left approximate identity[=LBAI], and let B be a Banach A − bimodule. Suppose that AB∗∗ and B∗∗A aresubset of B. Then if H1(A, B∗) = 0, it follows that H1(A, B(2n+1)) = 0.

Example 2.7. Assume that G is a compact group. Theni) we know that L1(G) is M(G) − bimodule and L1(G) is an ideal in the second dual ofM(G), M(G)∗∗. By using [19, corollary 1.2], we have H1(L1(G),M(G)) = 0. Then forevery n ≥ 1, by using preceding corollary, we conclude that

H1(L1(G),M(G)(2n+1)) = 0.

ii) we have L1(G) is an ideal in its second dual , L1(G)∗∗. By using [16], we know thatL1(G) is a weakly amenable. Then by preceding corollary, L1(G) is (2n + 1) − weaklyamenable.

Corollary 2.8. Let A be a Banach algebra and let A[2n] be dense in A. Suppose thatAB∗∗ and B∗∗A are subset of B. Then the following are equivalent.1. H1(A, B∗) = 0.2. H1(A, B(2n+1)) = 0 for some n ≥ 0.3. H1(A, B(2n+1)) = 0 for each n ≥ 0.

Corollary 2.9. [6]. Let A be a weakly amenable Banach algebra such that A is anideal in A∗∗. Then A is (2n + 1) − weakly amenable for each n ≥ 0.

Example 2.10. i) Let B be a dual Banach algebra and dimB < ∞. Then by usingProposition 2.6.24 from [5], we know that N(B), the collection of all operators fromB into B, is an ideal inN(B)∗∗. By using Corollary 4.3.6 from [24], N(B) is amenable,and so it also is weakly amenable. Consequently, by using the preceding corollary,N(B) is (2n + 1) − weakly amenable for every n ≥ 0.ii) We know that every von Neumann algebra A is an weakly amenable Banach algebra,see [24]. Now if A is an ideal in its second dual, A∗∗, then by using preceding corollary,A is (2n + 1) − weakly amenable Banach algebra for each n ≥ 0.Assume that A and B are Banach algebra. Then A ⊕ B , with norm

∥ (a, b) ∥=∥ a ∥ + ∥ b ∥,and product (a1, b1)(a2, b2) = (a1a2, b1b2) is a Banach algebra. It is clear that if X is aBanach A and B − bimodule, then X is a Banach A ⊕ B − bimodule.

92

K. Haghnejad Azar

Theorem 2.11. Suppose that A and B are Banach algebras. Let X be a BanachA and B − bimodule. Then, H1(A ⊕ B, X) = 0 if and only if H1(A, X) = H1(B, X) = 0.

Example 2.12. Let G be a locally compact group and X be a Banach L1(G)−bimodule.Then by [6, pp.27 and 28], X∗∗ is a Banach L1(G)∗∗ − bimodule. Since L1(G)∗∗ =LUC(G)∗ ⊕ LUC(G)⊥, by using preceding theorem, we have H1(L1(G)∗∗, X∗∗) = 0 ifand only if H1(LUC(G)∗, X∗∗) = H1(LUC(G)⊥, X∗∗) = 0.

Corollary 2.13. i) Suppose that A and B are Banach algebras. Let X be a BanachA and B − bimodule. Then A ⊕ B is an amenable Banach algebra if and only if A andB are amenable Banach algebras.ii) Let A be a Banach algebra and n ≥ 1. Then H1(⊕n

i=1A, A∗) = 0 if and only if A isweakly amenable.

Theorem 2.14. Let B be a Banach A − bimodule and suppose that every derivationfrom A into B∗ is weakly compact. If Zℓ

A∗∗(B∗∗) = B∗∗ and H1(A∗∗, B∗∗∗) = 0, then

H1(A, B∗) = 0.

Theorem 2.15. Let B be a Banach A − bimodule and suppose that D′′(A∗∗) ⊆ B∗. IfZℓ

A∗∗(B∗∗) = B∗∗ and H1(A∗∗, B∗∗∗) = 0, then H1(A, B∗) = 0.

Corollary 2.16. Let B be a Banach A − bimodule and let every derivationD : A→ B∗, satisfies D′′(A∗∗) ⊆ wapℓ(B). If H1(A∗∗, B∗∗∗) = 0, thenH1(A, B∗) = 0.

Corollary 2.17. Let B be a Banach A−bimodule and suppose that for every derivationD : A→ B∗, we have D′′(A∗∗)B∗∗ ⊆ A∗. If H1(A∗∗, B∗∗∗) = 0, then

H1(A, B∗) = 0.

Theorem 2.18. Let B be a Banach A − bimodule and suppose that AA∗∗ ⊆ A andB∗∗A = B∗∗. If H1(A∗∗, B∗∗∗) = 0, then H1(A, B∗) = 0.

[1] R. E. Arens, The adjoint of a bilinear operation, Proc. Amer. Math. Soc. 2 (1951), 839-848.[2] W. G. Bade, P.C. Curtis and H.G. Dales, Amenability and weak amenability for Beurling and

Lipschitz algebra, Proc. Lodon Math. Soc. 3 no.55 (1999) 359-377.[3] J. Baker, A. T. Lau, J.S. Pym Module homomorphism and topological centers associated with

weakly sequentially compact Banach algebras, Journal of Functional Analysis. 158 (1998), 186-208.

[4] F. F. Bonsall, J. Duncan, Complete normed algebras, Springer-Verlag, Berlin 1973.[5] H. G. Dales, Banach algebra and automatic continuity, Oxford 2000.[6] H. G. Dales, F. Ghahramani, N. Grønbæk Derivation into iterated duals of Banach algebras Studia

Math. 128 1 (1998), 19-53.[7] H. G. Dales, A. Rodrigues-Palacios, M.V. Velasco, The second transpose of a derivation, J.

London. Math. Soc. 2 64 (2001) 707-721.[8] N. Dunford, J. T. Schwartz, Linear operators.I, Wiley, New york 1958.[9] M. Eshaghi Gordji, M. Filali, Arens regularity of module actions, Studia Math. 181 3 (2007),

237-254.

93

Amenability

[10] M. Eshaghi Gordji, M. Filali, Weak amenability of the second dual of a Banach algebra, StudiaMath. 182 3 (2007), 205-213.

[11] F. Gourdeau, Amenability and the second dual of a Banach algebras, Studia. Math 125(1) (1997),75-81.

[12] F. Ghahramani, R. J. Loy and G. A. Willis, Amenability and weak amenability of second conjugateBanach algebras, Proc. American Math. Soc. 124 (1996), 1489-1496.

[13] B.E. Johoson, Cohomology in Banach algebra, Mem. Amer. Math. Soc. 127, 1972.[14] B. E. Johoson, Derivation from L1(G) into L1(G) and L∞(G), Harmonic analysis. Luxembourg

1987, 191-198 Lecture Note in Math., 1359, Springer, Berlin, 1988. MR 90a:46122.[15] B. E. Johoson, Weak amenability of group algebra, Bull. Lodon. Math. Soc. 23(1991), 281-284.[16] A. T. Lau, V. Losert, On the second Conjugate Algebra of locally compact groups, J. London

Math. Soc. 37 (2)(1988), 464-480.[17] A. T. Lau, A. Ülger, Topological center of certain dual algebras, Trans. Amer. Math. Soc. 348

(1996), 1191-1212.[18] V. Losert, The derivation problem for group algebra, Annals of Mathematics, 168 (2008), 221-

246.[19] S. Mohamadzadih, H. R. E. Vishki, Arens regularity of module actions and the second adjoint of

a derivation, Bulletin of the Australian Mathematical Society 77 (2008), 465-476.[20] M. Neufang, Solution to a conjecture by Hofmeier-Wittstock, Journal of Functional Analysis. 217

(2004), 171-180.[21] M. Neufang, On a conjecture by Ghahramani-Lau and related problem concerning topological

center, Journal of Functional Analysis. 224 (2005), 217-229.[22] V. Runde, Lectures on the amenability, springer-verlag Berlin Heideberg NewYork.[23] A. Ülger, Some stability properties of Arens regular bilinear operators, Proc. Amer. Math. Soc.

(1991) 34, 443-454.[24] A. Ülger, Arens regularity of weakly sequentialy compact Banach algebras, Proc. Amer. Math.

Soc. 127 (11) (1999), 3221-3227.[25] P. K. Wong, The second conjugate algebras of Banach algebras, J. Math. Sci. 17 (1) (1994),

15-18.[26] Y. Zhang, Weak amenability of a class of Banach algebra, Cand. Math. Bull. 44 (4) (2001) 504-

508.[27] Y. Zhang, Weak amenability of module extentions of Banach algebras, Trans. Amer. Math. Soc.

354 (10) (2002), 4131-4151.

K. Haghnejad Azar,Department of Mathematics-University of Mohaghegh [email protected]

94



January 20–21, 2016

Multi-bounded sets and projectivity of certain Banach modules

F. Hamidi Dastjerdi and S. Soltani Renani∗

Abstract

Let A be a Lau algebras. We obtain an equivalent condition for projectivity of certain Banach left A-modules in terms of multi-bounded sets. We then apply this result for certain Lau algebras on a locallycompact group.

2010 Mathematics subject classification: Primary 43A07, Secondary 46H05.Keywords and phrases: Multi-norms, multi-bounded set, Lau algebras, projectivity.

1. Introduction and preliminaries

LetA be a Banach algebra and P be a Banach leftA-module. Then the spaceA⊗P isa Banach leftA-module with the action a · (b⊗ ξ) = ab ⊗ ξ for all a, b ∈ A and ξ ∈ P.Define the leftA-module morphism π : A⊗P −→ P by the formula

π(a ⊗ ξ) = a · ξ

for ξ ∈ P and a ∈ A. It is shown in [7] that if P is essential as Banach left A-module(i.e., the linear span of A · P is dense in P), then P is projective if and only if thecanonical morphism π is a retraction.

The concept of projectivity of Banach modules was introduced and studied byHelemskii [7]. Projectivity of several Banach modules over group algebras have beenstudied by Dales and Polyakov [5] and by Ramsden [13]. Several authors have studiedhomological properties of some Banach modules; see, e.g. [1] and [14]. Nasr-isfahaniand second author [10] investigated homological properties of some Banach modulesrelated to the characters.

The theory of multi-norms spaces was first introduced and studied by Dales andPolyakov in [4]. This study of multi-norms was continued in [2] to resolve a long-standing question of Barry Johnson. Dales and polyakov characterized homologicalproperties of some Banach modules by multi-normed space [4]. Also, the theory ofmulti-norms has been studied by several authors; see for example [3] and [12].

In this paper, we study the relation between the multi-bounded sets and projectivityof certain Banach leftA-modules.∗ speaker

95

F. Hamidi Dastjerdi and S. Soltani Renani

2. Main results

We first recall the definition of a multi-normed space. Let E be a Banach space anddenote by En the n-fold Cartesian product of E.

A multi-norm based on En : n ∈ N is a sequence (∥.∥n) = (∥.∥n : n ∈ N) such that∥.∥n is a norm on En, such that ∥x∥1 = ∥x∥ for each x ∈ E, and such that the followingAxioms (A1) − (A4) are satisfied for each n ∈ N and x1, . . . , xn ∈ E:

(A1) ∥(xσ(1), . . . , xσ(n))∥n = ∥(x1, . . . , xn)∥n ∀ permutation σ;

(A2) ∥(α1x1, . . . , αnxn)∥n ≤ (max1≤i≤n

| αi |)∥(x1, . . . , xn)∥n (α1, . . . , αn ∈ C);

(A3) ∥(x1, . . . , xn−1, 0)∥n = ∥(x1, . . . , xn−1)∥n−1;

(A4) ∥(x1, . . . , xn−1, xn−1)∥n = ∥(x1, . . . , xn−1)∥n−1.

In this case, we say that ((En, ∥.∥n) : n ∈ N) is a multi-Banach space.

Example 2.1. Let (∥.∥αn : n ∈ N) : α ∈ A be the (non-empty) family of all multi-norms on En : n ∈ N for each α in a non-empty index set A (perhaps finite). Forn ∈ N, set

∥|(x1, . . . , xn)|∥n := supα∈A∥|(x1, . . . , xn)|∥αn (x1, . . . , xn ∈ E).

Then (∥|.|∥ : n ∈ N) is a multi-norm on En : n ∈ N, called the maximum multi-norm.hence we have for each n ∈ N and x = (x1, . . . , xn) ∈ En,

∥x∥maxn = sup

|

n∑i=1

⟨xi, λi⟩ |: λ1, . . . , λn ∈ E′, µ1,n(λ1, . . . , λn) ≤ 1,

where the supremum is take over all λ1, . . . , λn in the dual En and µ1,n is the weak1−summing norm.

Let ((En, ∥.∥n) : n ∈ N) be a multi-normed space. A subset F ⊂ E is multi-boundedif

mb(F) := sup∥(x1, . . . , xn)∥maxn : x1, . . . , xn ∈ F, n ∈ N < ∞;

the constant mb(F) is the multi-bound of F.

A Banach algebraA is called Lau algebra if the dual spaceA′ ofA is a W∗-algebraand the identity element u of A′ is a multiplicative linear functional on A. Examplesof Lau algebras include the group algebra and the Fourier algebra of a locally compactgroup and quantum group algebras, or more generally the predual algebra of a Hopfvon Neumann algebra.

LetA be a Lau algebra and let P be a Banach space. We denote by Pu the BanachleftA-module endowed with the action defined by

96

Multi-bounded sets and projectivity

a · ξ = u(a)ξ (ξ ∈ P, a ∈ A).

Note that the Banach leftA-module Pu is essential.We are now ready to prove our main result.

Theorem 2.2. Let A be a Lau algebra and let u be the identity element of A′. Thenthe following statements are equivalent:

(a) Any Banach leftA-module Pu is projective;(b) The Banach leftA-module Cu is projective;(c) There exists an element m ∈ P1(A) such that the set aim : ai ∈ P1(A) is

multi-bounded.

Proof. The equivalence (a)⇔ (b) is trivial by Theorem 4.3 of [10].(b)⇒ (c). Suppose that Cu is a projective Banach leftA-module with a · z = u(a)z

for all z ∈ Cu and a ∈ A. Then there exists

ρ : Cu → A⊗Cu ≃ A

such that (π ρ)(z) = z for all z ∈ Cu, where π : A⊗Cu → Cu is canonical embedding.Define m = ρ(1) ∈ A. Then we have

supn∈N

∥(a1m, . . . , anm)∥max

n : a1, . . . , an ∈ A= sup

n∈N

∥(ρ(a1 · 1), . . . , ρ(an · 1))∥max

n

= sup

n∈N

∥u(a1)m, . . . , u(an)m∥max

n

= sup

n∈N

∥m, . . . ,m∥max

n

= ∥m∥.

This shows that, (b) holds.(c) ⇒ (b). Suppose that there is an element m ∈ P1(A) such that the set aim :

ai ∈ P1(A) is multi-bounded. Then by [2, Theorem 5.7], the set aim : ai ∈ P1(A) isrelatively weakly compact and also, by the Krein − S mulian Theorem, the followingset is weakly compact

K := co(aim) = n∑

i=1

tiaim : 0 ≤ ti ≤ 1,n∑

i=1

ti = 1, ai ∈ P1(A).

Now, let Σ = Lb : b ∈ P1(A) be a semigroup of isometric affine maps from theweakly compact convex set K into itself defined by

Lb(Λ) = bΛ (b ∈ P1(A),Λ ∈ K).

Hence, by the Ryll−Nardzewski fixed point Theorem, given in [6, 11], there existsm0 ∈ K which is a common fixed point for the set Σ such that Lb(m0) = m0. Then for

97

F. Hamidi Dastjerdi and S. Soltani Renani

Banach left A-module Cu with a · z = u(a)z for all z ∈ Cu and a ∈ A, we define themap ρ : Cu → A⊗Cu ≃ A with

ρ(z) = m0 ⊗ z (z ∈ Cu)

that has the following properties:

ρ(a · z) = m0 ⊗ a · z = u(a)m0 ⊗ z = am0 ⊗ z = a · ρ(z),

andπ ρ(z) = π(m0 ⊗ z) = m0 · z = u(m0)z = z.

Therefore, ρ is clearly a right inverse to π, and so Cu is a projective Banach left A-module.

Let G be a locally compact group with left Haar measure λG and let L1(G) =L1(G, λG) be the group algebra of G as defined in [8] endowed with the norm ∥.∥1 andthe convolution product ∗ given by

(φ ∗ ψ)(s) =∫

Gφ(t)ψ(t−1s)dλG(t) (φ, ψ ∈ L1(G), s ∈ G),

As a consequence of Theorem 4.3 and Theorem 6.1 of [10], we have the followingresult.

Corollary 2.3. Let G be a locally compact group. Then the following statements areequivalent:

(a) G is compact;

(b) The Banach left L1(G)-module Cu is projective.

We have the following consequence of Theorem 2.2 and Corollary 2.3.

Corollary 2.4. Let G be a locally compact group. Then the following statements areequivalent:(a) G is compact;

(b) There is an element m ∈ P1(L1(G)) such that mbs1 ·m, . . . , sn ·m : si ∈ G

< ∞

This result was also obtained by Ramsden [12, Proposition 4.2].

[1] F. Bahrami., R. Nasr-Isfahani., and S. Soltani Renani., Homological properties of certain Banachmodules over group algebras, Proc. Edinb. Math. Soc. 54 (2011), No. 2, 321-328.

[2] H. G. Dales., M. Daws., H. L. Pham., and P. Ramsden., Multi-norms and the injectivity of Lp(G),J. London. Math. Soc. 86 (2012), No. 3, 779-809.

[3] H. G. Dales., N. J. Laustsen., T. Oikhberg., and V. Troitsky., Multi-norms and Banach lattices,pp. 100, in preparation.

98

Multi-bounded sets and projectivity

[4] H. G. Dales., andM. E. Polyakov., Multi-normed spaces, Diss. Math. 488 (2012), 1-165.[5] H. G. Dales., andM. E. Polyakov., Homological properties of modules over group algebras, Proc.

London. Math. Soc. 89 (2004), No. 3, 390-426.[6] F. P., Greeleaf., Invariant means on topological groups, van Nostrand Mathematical Studies, 16

(van Nostrand Reinhold, New York, 1969).[7] A. Ya. Helemskii., The homology of Banach and topological algebras, Kluwer Academic

Publishers, Dordrecht, (1986).[8] E. Hewitt., and K. Ross., Abstract harmonic analysis II, Springer, New York, (1970).[9] A.T. LAU., Analysis on a class of Banach algebras with applications to harmonic analysis on

locally compact groups and semigroups, Fund. Math. 118 (1983), 161-175.[10] R. Nasr Isfahani., and S. Soltani Renani., Character contractibility of Banach algebras and

homological properties of Banach modules, Studia Math. 202 (2011), 205-225.[11] A. L. T., Paterson., Amenability, Amer. Math. Soc, Providence, Rhode Island, (1988).[12] P. Ramsden., Multi-norms and modules over group algebras, preprint, (2009).[13] P. Ramsden., Homological properties of modules over semigroup algebras, J. Funct Anal. 258

(2010), 3988-4009.[14] M. C. White., Injective module for uniform algebras, Proc. London Math. Soc. 73 (1996), 155-

184.

F. Hamidi Dastjerdi,Department of Mathematical Sciences,Isfahan University of Technology,Isfahan 84156-83111, Irane-mail: [email protected]

S. Soltani Renani,Department of Mathematical Sciences,Isfahan University of Technology,Isfahan 84156-83111, Irane-mail: [email protected]

99



January 20–21, 2016

The BSE property of certain Banach algebras

Z. Kamali∗

AbstractThe concepts of BSE property and BSE algebras were introduced and studied by Takahasi and Hatori in1990 and later by Kaniuth and Ulger. This abbreviation refers to a famous theorem proved by Bochnerand Schoenberg for L1(R), where R is the additive group of real numbers, and by Eberlein for L1(G)of a locally compact abelian group G. In this paper we investigate this property for the Banach algebraLp(S , µ) (1 ≤ p < ∞) where S is a compact totally ordered semigroup with multiplication xy = maxx, yand µ is a regular bounded continuous measure on S . As an application, we have shown that L1(S , µ) isnot an ideal in its second dual.

2010 Mathematics subject classification: Primary 46Jxx, Secondary 22A20.Keywords and phrases: BSE algebra. Totally ordered semigroup.

1. Introduction

Let A be a commutative Banach algebra. Denote by ∆(A) and M(A) the Gelfandspectrum and the multiplier algebra of A, respectively. A bounded continuous functionσ on ∆(A) is called a BSE-function if there exists a constant C > 0 such that for everyfinite number of φ1, ..., φn in ∆(A) and complex numbers c1, ..., cn, the inequality∣∣∣∣∣∣∣∣

n∑j=1

c jσ(φ j)

∣∣∣∣∣∣∣∣ ≤ C .

∥∥∥∥∥∥∥∥n∑

j=1

c jφ j

∥∥∥∥∥∥∥∥A∗

holds. The BSE-norm of σ (∥σ∥BS E) is defined to be the infimum of all such C. Theset of all BSE-functions is denoted by CBS E(∆(A)). Takahasi and Hatori [12] showedthat under the norm ∥.∥BS E , CBS E(∆(A)) is a commutative semisimple Banach algebra.

A bounded linear operator on A is called a multiplier if it satisfies xT (y) = T (xy)for all x, y ∈ A. The setM(A) of all multipliers of A is a unital commutative Banachalgebra, called the multiplier algebra of A.

For each T ∈ M(A) there exists a unique bounded continuous function T on ∆(A)such that T (a)(φ) = T (φ)a(φ) for all a ∈ A and φ ∈ ∆(A). See [7] for a proof.

DefineM(A) = T : T ∈ M(A).

∗ speaker

100

Z. Kamali

A bounded net (eα)α in a Banach algebra A is called a ∆-weak bounded approxi-mate identity for A if φ(eα) → 1 (equivalently, φ(eαa) → φ(a) for every a ∈ A) for allφ ∈ ∆(A). As is shown in [12], A has a ∆-weak bounded approximate identity if andonly if M(A) ⊆ CBS E(∆(A)).

A commutative Banach algebra A is called without order if aA = 0 implies a = 0(a ∈ A).

A commutative and without order Banach algebra A is called a BSE-algebra (orhas BSE-property) if it satisfies the condition

CBS E(∆(A)) = M(A).

The abbreviation BSE stands for Bochner-Schoenberg-Eberlein and refers to afamous theorem, proved by Bochner and Schoenberg [2, 11] for the additive groupof real numbers and in general by Eberlein [4] for a locally compact abelian group G,saying that, in the above terminology, the group algebra L1(G) is a BSE-algebra (See[8] for a proof).

It is worth to note that the semigroup algebra l1(Z+) (where Z+ is the additivesemigroup of nonnegative integers) is a BSE algebra [13], but for k ≥ 1, l1(Nk)(Nk = k, k + 1, k + 2, ...) is not a BSE algebra.

In [6], we established affirmatively a question raised by Takahasi and Hatori [12]that whether L1(R+) is a BSE-algebra.

The aim of the present paper is to show that for any totally ordered compactsemigroup S with multiplication xy = maxx, y and a regular bounded continuousmeasure µ on S , Lp(S , µ) (1 ≤ p < ∞) is not a BSE algebra. However, for anycompact abelian group G and 1 ≤ p < ∞, the Banach algebra Lp(G) is BSE [14].

As an application, we will show that the Banach L1(S , µ) is not an ideal in itssecond dual. However, for a locally compact group G, the Banach algebra L1(G) is anideal in its second dual if and only if G is compact.

Finally, we prove that CBS E(n)((0, 1]) = Cb((0, 1]) for any natural number n.

2. BSE property of totally ordered semigroup algebras

The Banach algebra Lp(S , µ) (1 ≤ p < ∞), whenever S is a totally ordered compactspace with a regular bounded continuous measure µ on S , was first introduced bySapounakis [9, 10] and then extensively studied more by Baker, Pym and Vasudeva[1].

Recall that a totally ordered locally compact space S is a totally ordered spaceS which is locally compact in its order topology. This has a natural continuousmultiplication xy = maxx, y. For convenience, we adjoin a minimal element 0 toS if it has not already got one. The convolution product v1 ∗ v2 of two bounded regularmeasures v1, v2 on S is defined (as a linear functional on the space C0(S ) of continuousfunctions vanishing at infinity on S ) by the usual formula∫

f dv1 ∗ v2 =

∫ ∫f (xy)dv1(x)dv2(y) ( f ∈ C0(S )).

101


By dividing the range of the inner integral on the right into the sets where x < y andx ≥ y we have

dv1 ∗ v2(x) = v1[0, x[ . dv2(x) + v2[0, x[ . dv1(x).

In particular, if both v1 and v2 are absolutely continuous with respect to some positivemeasure µ, say dv1 = f dµ and dv2 = gdµ, then so is v1 ∗ v2; and if we putdv1 ∗ v2 = f ∗ g.dµ, then

f ∗ g(x) = g(x)∫

[0,x[f dµ + f (x)

∫[0,x]

gdµ. (III)

By defining the convolution of two measurable functions f and g with respect to µ asin (III), one has the following result.

Proposition 2.1. For p = 1, Lp(S , µ) is a Banach algebra. For 1 < p ≤ ∞, Lp(S , µ) isa Banach algebra if and only if µ is bounded. Moreover, for 1 ≤ p ≤ ∞ the algebraLp(S , µ) is commutative and semisimple. It has an approximate identity if 1 ≤ p < ∞,which is bounded if and only if p = 1.

Proof. See [1].

The Cantor function. The function φ : [0, 1]→ [0, 1] defined by

φ(x) =∞∑

k=1

txk/22k ,

whenever the digit 1 does not appear in the ternary expansion of x, and

φ(x) =jx−1∑k=1

txk/22k +

12 jx

,

whenever the digit 1 does appear in the ternary expansion of x, and jx = mink : txk =

1, is called the Cantor funtion.

Proposition 2.2. For the Cantor function φ the following statements are valid.

1. φ is continuous and of bounded variation.2. φ is not absolutely continuous.3. φ is differentiable almost every where and φ′ = 0 a. e. on [0, 1].4.

∫ 10 φ(t)dt = 1

2 .

Proof. See [3] and [5].

102

Z. Kamali

Lemma 2.3. Let φ be the Cantor function. Then the function g defined by g(x) = xφ(x)is not absolutely continuous on [0, 1].

The following theorem is indeed the main result of this paper.

Theorem 2.4. The Banach algebra Lp(S , µ)(1 ≤ p < ∞) is not a BSE algebra.

It is well known that for a locally compact group G, the Banach algebra L1(G) isan ideal in its second dual if and only if G is compact. As a consequence of the abovetheorem, in the following result, we prove that this is not the case for the compacttotally ordered semigroup S .

Corollary 2.5. The Banach algebra L1(S , µ) is not an ideal in its second dual.

In the sequel we will prove that for any n ∈ N, the natural numbers, CBS E(n)((0, 1]) =Cb((0, 1]), where CBS E(n)(∆(A)) denotes the set of all complex valued continuous func-tions σ on ∆(A) which satisfy the following condition: there exists a positive realnumbers β such that the inequality∣∣∣∣∣∣∣∣

n∑j=1

c jσ(φ j)

∣∣∣∣∣∣∣∣ ≤ β .∥∥∥∥∥∥∥∥

n∑j=1

c jφ j

∥∥∥∥∥∥∥∥A∗

holds for any choice of complex numbers c1, ..., cn and φ1, ..., φn ∈ ∆(A). For eachσ ∈ CBS E(n)(∆(A)) we denote by ∥σ∥BS E(n) the infimum of such β. Let CBS E(∞)(∆(A)) =∩n∈N

CBS E(n)(∆(A). It is evident that ∥σ∥BS E = supn∈N ∥σ∥BS E(n) and

CBS E(∆(A)) = σ ∈ CBS E(∞)(∆(A) : ∥σ∥BS E < ∞.

Also we have

A ⊆ CBS E(∆(A)) ⊆ CBS E(∞)(∆(A))⊆ ... ⊆ CBS E(2)(∆(A)) ⊆ CBS E(1)(∆(A)) = Cb(∆(A)).

For more details see [13].

Theorem 2.6. For S = ([0, 1],max) and n ∈ N, we have

L1(S , µ) & CBS E((0, 1]) & CBS E(n)((0, 1]) = Cb((0, 1]).

Conjecture. It is well known that L∞(S , µ) with the pointwise multiplication is aC∗−algebra and so it is a BSE algebra. We conjecture that L∞(S , µ) with the convolu-tion multiplication is not a BSE algebra.

103


[1] Baker, J. W., Pym, J.S. and Vasudea, H. L.: Totally ordered measure spaces and their Lp algebras,Mathematika, 29, 42-54 (1982)

[2] Bochner, S.: A theorem on Fourier- Stieltjes integrals, Bull. Amer. Math. Soc, 40, 271-276 (1934)[3] Dovgoshey, O., Martio, O., Ryazanov, V. and Vuorinen, M.: The Cantor function, Expo. Math.

24, 1-37 (2006)[4] Eberlein, W. F.: Characterizations of Fourier-Stieltjes transforms, Duke Math. J. 22, 465-468

(1955)[5] Gorin, E. A., Kukushkin, B. N.: Integrals Related to the Cantor Function. St. Petersburg Math. J.

15, 449-468 (2004)[6] Kamali, Z., Lashkarizadeh Bami, M.: The Bochner-Schoenberg-Eberlein property for L1(R+), J.

Fourier Analysis and Applications 20, 225-233 (2014)[7] Larsen. R.: An introduction to the theory of multipliers, Springer-Verlag, New York, 1971[8] Rudin, W.: Fourier analysis on groups, Wiley Interscience, New York, 1984[9] Sapounakis, A.: Properties of measures on topological spaces, Thesis, University of Liverpool,

1980[10] Sapounakis, A.: Measures on totally ordered spaces. Mathematika 27, 225-235 (1980)[11] Schoenberg, I. J.: A remark on the preceding note by Bochner, Bull. Amer. Math. Soc. 40, 277-

278 (1934)[12] Takahasi, S. -E., Hatori, O.: Commutative Banach algebras which satisfy a Bochner-Schoenberg-

Eberlein-type theorem, Proc. Amer. Math. Soc. 110, 149-158 (1990)[13] Takahasi, S. -E., Hatori, O.: Commutative Banach algebras and BSE-inequalities, Math. Japonica

37, 47-52 (1992)[14] Takahasi, S. -E. , Takahashi, Y., Hatori,O and Tanahashi, K.: Commutative Banach algebras and

BSE-norm, Math. Japonica 46 , 273-277 (1997)

Z. Kamali,Department of Mathematics,Isfahan (Khorasgan) Branch,Islamic Azad University,Isfahan, Irane-mail: [email protected]

104



January 20–21, 2016

On the weak∗ continuity of LUC(G)∗-module action onLUC(X,G)∗ related to G-space X

H. Javanshiri∗ and N. Tavallaei

Abstract

For a locally compact group G and a G-space X, we use the notation LUC(X,G) to denote the Banachspace of all bounded left uniformly continuous functions on X. We introduce a left action of LUC(G)∗

on LUC(X,G)∗ to make it a Banach left module and then we study the topological centre related to thismodule action. We propound some main properties of that topological centre, extend the main results oftopological centre related to a locally compact group to a G-space, and then we apply our results to somespecial homogeneous spaces.

2010 Mathematics subject classification: Primary 43A85, Secondary 46H25.Keywords and phrases: Locally compact group, G-space, topological centre, left uniformly continuousfunction, Banach left module, measure algebra.

1. Introduction

Let G be a locally compact group. Then the Banach space LUC(G)∗, the topologicaldual of the space of bounded left uniformly continuous functions on G, is a Banachalgebra equipped with the first Arens product "⊙”. In general, this product is notseparately weak∗ to weak∗ continuous on LUC(G)∗, and in recent years there hasbeen shown considerable interest by harmonic analysts in the characterization of thefollowing space

Z(G) =m ∈ LUC(G)∗ : n 7→ m ⊙ n is weak∗ to weak∗ continuous

.

As far as we know the subject, the starting point of the study of the space Z(G) is thepaper by Zappa [3]. In this paper Zappa proved that Z(R) is precisely M(R), where Ris the additive group of real number and M(R) is the Banach algebra of all complexRadon measures on R. This result was extended to all abelian locally campact groupsby Grosser and Losert in [1], and to all locally compact groups by Lau in [2].

In this paper, considering X as a locally compact Hausdorff space on which Gacts continuously from the left, we introduce the Banach space LUC(X,G) as wellLUC(G). Then we present a left action of LUC(G)∗ on LUC(X,G)∗, as an extension∗ speaker

105

H. Javanshiri and N. Tavallaei

of the natural action of M(G) on M(X), to make a Banach left LUC(G)∗-module. Thenwe investigate the topological centre related to this module action and extend the mainresults of Lau [2] from locally compact groups to G-spaces.

2. The definitions and some basic results

Throughout this paper, G is a locally compact group. A locally compact Hausdorffspace X is a G-space if there is a continuous action map G × X → X, denoted by(s, x) 7→ s · x, satisfying (st) · x = s · (t · x) and e · x = x, for all s, t ∈ G and x ∈ X. Theaction of G on X is said to be effective, when the following closed normal subgroup ofG is trivial

N(X,G) =s ∈ G : s · x = x for all x ∈ X

.

Moreover, if M(X) denotes the Banach space of all complex Radon measures on Xwith total variation norm, then the action of G on X induces an action of M(G) onM(X) which makes M(X) as a Banach left M(G)-module. The left translation of afunction F ∈ Cb(X) defined by lsF(y) = F(s · y) for all y ∈ X and we consider

LUC(X,G) =F ∈ Cb(X) : s ∈ G 7→ lsF ∈ (Cb(X), ∥ · ∥∞) is continuous

,

which is a closed subspace of Cb(X). Moreover, if LUC(X,G)∗ denotes the first dualspace of the Banach space LUC(X,G), M is an arbitrary element of LUC(X,G)∗ andF ∈ LUC(X,G), we define the function MF : G → C by MF(s) = ⟨M, lsF⟩, for alls ∈ G, which belongs to LUC(G). Also, the mapping

LUC(X,G)∗ × LUC(X,G)→ LUC(G)(M, F) 7→ MF

is a bounded bilinear map with ∥MF∥∞ ≤ ∥M∥ ∥F∥∞. So, we can define a boundedbilinear map as follows

LUC(G)∗ × LUC(X,G)∗ → LUC(X,G)∗

(m,M) 7→ m · M

with ∥m · M∥ ≤ ∥m∥ ∥M∥, where ⟨m · M, F⟩ = ⟨m, MF⟩, for all F ∈ LUC(X,G).

Proposition 2.1. Let G be a locally compact group and X be a G-space. Under themapping (m,M) 7→ m · M, LUC(X,G)∗ becomes a Banach left LUC(G)∗-module with∥m · M∥ ≤ ∥m∥ ∥M∥ and δe · M = M.

If now, for a given σ ∈ M(X), we define a linear functional on LUC(X,G), denotedagain by σ, which assigns to each F ∈ LUC(X,G) the value

∫X F(x) dσ(x). Then,

M(X) may be regarded as a subspace of LUC(X,G)∗. Moreover, it is not hard tocheck that the inclusion LUC(G)∗ · M(X) ⊆ M(X) can fail even if X = G, where

LUC(G)∗ · M(X) =m · σ : m ∈ LUC(G)∗, σ ∈ M(X)

;

In other word, the Banach space M(X) is not in general an LUC(G)∗-submodule ofLUC(X,G)∗. On the other hand, if X is compact, then M(X) is a LUC(G)∗-submodule

106

On the weak∗ continuity of LUC(G)∗-module LUC(X,G)∗

of LUC(X,G)∗; This is because of, in this case M(X) = LUC(X,G)∗. We do notknow if the converse of this fact is valid in general; here, we prove the converse underan extra assumption. The notation CLS(X,G) in this proposition and in the sequeldenotes the norm closure of the linear span of the set

LUC(X,G)∗LUC(X,G) :=MF : M ∈ LUC(X,G)∗, F ∈ LUC(X,G)

,

with respect to the norm topology of LUC(G). Also, we say that the action of LUC(G)∗

on LUC(X,G)∗ is faithful if m ∈ LUC(G)∗ is so that m·M = 0 for all M ∈ LUC(X,G)∗,then m = 0.

Proposition 2.2. Let G be a locally compact group and X be a G-space. Then thefollowing statements hold.

(a) CLS(X,G) = LUC(G) if and only if the action of LUC(G)∗ on LUC(X,G)∗

is faithful.

(b) If the action of LUC(G)∗ on LUC(X,G)∗ is faithful, then G acts effectively onX.

(c) If X is compact, then M(X) is an LUC(G)∗-submodule of LUC(X,G)∗. Theconverse is also true if X is a transitive G-space.

3. The weak∗ continuity of the left LUC(G)∗-module action

A problem which is of interest is that for which element m ∈ LUC(G) the mapM 7→ m · M on LUC(X,G)∗ is weak∗ to weak∗ continuous? Therefore, it seemsvaluable to define

Z(X,G) =m ∈ LUC(G)∗ : M 7→ m · M is weak∗ to weak∗ continuous on LUC(X,G)∗

,

the topological centre of the module action induced by LUC(G)∗ on LUC(X,G)∗.In the special case that we let G act on itself by left multiplication, the set Z(G,G)coincides with Z(G). This section studies the subspace Z(X,G) of LUC(G)∗ in thecase where X is a G-space and, in particular, the question when the subspace Z(X,G)is M(G). Before proceeding further in this section, we should note that if X is a G-space, then LUC(X,G)∗ is a left Banach G-module; In fact, it is suffices to define a leftaction of G on LUC(X,G)∗ by (s,M) 7→ δs · M.

Now, let X be a G-space. Given F ∈ LUC(X,G) and x ∈ X, we define rxF onG by (rxF)(s) = F(s · x),s ∈ G. A routine computation shows that rxF is a functionin LUC(G). Hence, if m is an arbitrary element of LUC(G)∗, then we can define acomplex-valued function Fm on X by Fm(x) = ⟨m, rxF⟩, x ∈ X. Obviously thatFm is a bounded function on X with ∥Fm∥∞ ≤ ∥m∥ ∥F∥∞. Moreover, suppose thatm is an element of LUC(G)∗ such that for each F ∈ LUC(X,G) the function Fmis in LUC(X,G), then every M in LUC(X,G)∗ gives a linear functional M • m onLUC(X,G)∗ by ⟨M •m, F⟩ = ⟨M, Fm⟩. One may easily check that, if M •m = m · M,for all M ∈ LUC(X,G)∗, then m ∈ Z(X,G).

The following results, gives some of the main properties of the set Z(X,G).

107

H. Javanshiri and N. Tavallaei

Proposition 3.1. Let G be a locally compact group and X be a G-space. Then thefollowing assertions hold.

(a) Z(X,G) is a closed subalgebra of LUC(G)∗.

(b) M(G) is contained in Z(X,G).

Theorem 3.2. Let G be a locally compact non-compact group and let X be a G-space.Then the following assertions are equivalent.

(a) Z(X,G) = M(G).

(b) The action of LUC(G)∗ on LUC(X,G)∗ is faithful.

Example 3.3. Let G be a locally compact group and X be a G-space.

(a) If G is compact, then Z(X,G) = M(G).

(b) If X is a finite discrete space, then Z(X,G) = LUC(G)∗. In particular, if G isnon-compact, then M(G) ⫋ Z(X,G) = LUC(G)∗.

Example 3.4. Let G be a locally compact, non-compact group and let H and K aretwo non-trivial closed normal subgroup of G.

(a) If X = G/H , then M(G) ⊊ Z(X,G). In particular, if the index of H in G isfinite, then Z(X,G) = LUC(G)∗.

(b) If X = K , then, obviously, G acts on X by conjugation. In this case,N(X,G) = CG(K). Hence, we can say if either the centralizer CG(G) or CG(K)is non-trivial (for example, if K is abelian), then we have M(G) ⊊ Z(X,G).

[1] M. Grosser, V. Losert, The norm-strict bidual of a Banach algebra and the dual of Cu(G)∗,Manuscripta Math. 45 (1994), no. 2, 127–146.

[2] A. T.-M. Lau, Continuity of Arens multiplication on the dual space of bounded uniformly contin-uous functions on locally compact groups and topological semigroups, Math. Proc. CambridgePhilos. Soc. 99 (1986), no. 02, Cambridge University Press, pp. 273–283.

[3] A. Zappa, The centre of the convolution algebra Cu(G)∗, Rend. Sem. Mat. Univ. Padova 52 (1975),71–83.

H. Javanshiri,Department of Mathematics, Yazd University, Yazd, Irane-mail: [email protected]

N. Tavallaei,Department of Mathematics, School of Mathematics and Computer Science, Univer-sity of Damghan, Damghan, Irane-mail: [email protected]

108



January 20–21, 2016

The Second Dual of Certain Triple Systems

A. A. Khosravi

Abstract

In this talk, by mimicking the Arens’ method, we extend the triple product of a JB∗−triple system M tosome triple products on the second dual M∗∗ of M. We then show that all these triple products on thesecond dual of a JB∗−triple system M coincide. We further show that M∗∗ is a JBW∗−triple system. Thecoincidence of triple products on the second dual of L1(G), where G is a locally compact group, is alsodiscussed.

2010 Mathematics subject classification: Primary 17C65, Secondary 46H25..Keywords and phrases: JB∗-triple, Arens product, second dual, convolution algebra.


JB∗−triples arose in the study of bounded symmetric domains in Banach spaces [7].A JB∗−triple system is a complex Banach space M together with a continuous tripleproduct ·, ·, · : M × M × M → M satisfying the following conditions:

(1) ·, ·, · is symmetric and bilinear in the outer two variables and conjugate-linearin middle variable;

(2) ·, ·, · obeys the so-called Jordan identity

a, b, x, y, z = a, b, x, y, z − x, b, a, y, z + x, ya, b, z, orequivalently,

ϕ(a, b)ϕ(x, y) − ϕ(x, y)ϕ(a, b) = ϕ(ϕ(a, b)x, y) − ϕ(x, ϕ(b, a)y),

for all a, b, x, y, z ∈ M, where ϕ(a, b)(x) := a, b, x;

(3) for each a ∈ M, the operator ϕ(a, a) from M to M is a hermitian operator withnon-negative spectrum;

(4)∥x, x, x| = ∥x∥3 for each x ∈ M.

109

A. A. Khosravi

Note that ϕ(a, b) is sometimes denoted by ab and operators of this form are calledbox operators.

C∗−algebras and JB∗−algebras are the main examples of JB∗−triples. Moreprecisely, every C∗−algebra (resp. every JB∗−algebra) is a JB∗−triple with respect tothe product x, y, z = 1

2 (xy∗z + zy∗x) (resp. x, y, z = (xoy∗)oz + (zoy∗)ox − (xoz)oy∗).The basic facts about JB∗−triples can be found in [7] and some of the referencestherein.

A JBW∗−triple system is a JB∗−triple which is also a dual Banach space (witha unique isometric predual). It is known that the triple product of a JBW∗−triple isseparately weak∗ continuous. S. Dineen [2], by the ultrapower techniques, showedthat the second dual M∗∗ of every JB∗−triple M is a JB∗−triple itself (see also [4]).Our main aim is to show that every JB∗−triple system M is (ternary) Arens regular;that is, all triple products on M∗∗, induced by the original triple product ·, ·, ·, arecoincide and that M∗∗ is a JB∗−triple system itself. We also discuss the coincidenceof these triple products on the second dual of the group convolution ternary algebraL1(G). For ample information with more details one may consult to [6].

2. Main results

Let (M, ·, ·, ·) be a JB∗-triple system. Then, similar to Arens’ method [1], thereare six (Arens type) extensions of the triple product ·, ·, · to the second dual M∗∗, allof them are bilinear in the outer two variables and conjugate linear in the middle one.

In [5], Friedman and Russo proved a Gelfand-Naimark Theorem for JB∗−tripleswhich states: Every JB∗−triple is isometrically isomorphic to a subtriple of theJB∗−triple system B(H)

⊕∞C(S ,C6), where S is the Stone-Cech compactification

of a discrete set and C6 is the set of 3 × 3 hermitian matrices over the complex Cayleynumbers. Using this result we shall prove our main result as follows, which recovershe main results of [2] and [4].

Theorem 2.1. Every JB∗−triple system M is ternary Arens regular. Moreover, M∗∗ isa JBW∗−triple system.

As an immediate consequence, we arrive at the next well known result of Civinand Yood [3].

Theorem 2.2. Every C∗−algebra A is Arens regular and the second dual A∗∗ of A is avon Neumann algebra.

In the case where ·, ·, · : M × M × M → M is trilinear and symmetric in outervariables and satisfies the Jordan identity, then M is called a Jordan triple system. If·, ·, · is conjugate linear in middle variable then M is called a hermitian Jordan triplesystem.

Now we equip the group convolution algebra L1(G) with a hermitian jordan triplestructure by the triple product · · · : L1(G) × L1(G) × L1(G) → L1(G) defined by f , g, h = 1

2 ( f ⋆ g∗ ⋆ h + h ⋆ g∗ ⋆ f ).

110

The Second Dual of Certain Triple Systems

In the setting of Jordan triples, a commutative Joradan triple is the one whose boxoperators form a commutative set. That is, [ab, xy] = 0 for all a, b, x, y ∈ M. It canbe readily verified that, (L1(G), · · · ) is commutative if and only if G is abelian.

Note that L1(G) is not a JB∗-triple system, in general. For the ternary Arensregularity of L1(G) we present the following result, see [8].

Theorem 2.3. L1(G) is ternary Arens regular if and only if G is finite.

[1] R. Arens, The adjoint of a bilinear operation, Proc. Amer. Math. Soc. 2 (1951), 839-848.[2] S. Dineen, The second dual of a JB−triple system, Complex Analysis, Functional Analysis and

Approximation Theory (Ed. J. Mujica), North-Holland, Amsterdam, 1986.[3] P. Civin and B. Yood, The second conjugate space of a Banach algebra as an algebra, Pacific J.

Math. 11 (1961), 847-870.[4] J. M. Isidro, W. Kaup and A. Rodriguez, On real forms of JB∗−triples, Manuscripta Math. 86

(1995), 311-335.[5] Y. Friedman and B. Russo, The Gelfand-Naimark theorem for JB∗−triples, Duke Math. J. 53

(1986), 191-208.[6] A.A. Khosravi and H.R. Ebrahimi Vishki, The second conjugate space of a Banach ∗−triple

system, Preprint (2016).[7] B. Russo, Structure of JB−triples, Jordan Algebras,Proceedings of the Oberwolfach Conference

1992 (Ed. W. Kaup, K. Mc Crimmon and H. Petersson), de Gruyter, Berlin, 1994, 209280.[8] N.J. Young, Irregularity of multiplication in group algebras, Quart. J. Math. 24 (1973) 59-62.

A. A. Khosravi,Department of pure mathematics,Ferdowsi University, P.O. Box 1159,Mashhad 91775, IRAN.e-mail: [email protected]

111



January 20–21, 2016

Frames, operators and duality principle

A. Khosravi and F. Takhteh∗

Abstract

In this paper, for every Bessel sequence and every Riesz basis, we consider an anti-linear operator on H.We give important characterizations of frames, Riesz bases, Riesz sequences, and Besselian frames interms of this anti-linear map, then we consider the concept of R-duality with respect to Riesz bases andwe apply our results to characterizing of frames, Riesz bases, Riesz sequences in terms of their R-dualsequences with respect to Riesz bases.

2010 Mathematics subject classification: Primary 42C15, Secondary 42A38.Keywords and phrases: Frame, Ron-Shen duality, Riesz basis.

1. Introduction

Let g be a function in L2(R) and a, b two positive constants. The collectionEmbTnagm,n∈Z where Emb f (x) = e2πimbx f (x) and Tna f (x) = f (x − na) is called aGabor frame in L2(R) if it is a frame for the Hilbert space L2(R).One of the most important results in Gabor frame is Ron-Shen duality principle [8]that precisely characterizes Gabor frames. It states that for every g ∈ L2(R) anda, b > 0 with ab ≤ 1, EmbTnagm,n∈Z is a frame with bounds A, B for L2(R) if only if 1√

abE m

aT n

bgm,n∈Z is a Riesz sequence with bounds A, B.

For generalizing duality principles in Gabor frames to abstract frame theory, in [2]the authors introduced the concept of R-duality with respect to orthonormal bases asfollows:Let (e j) j∈N and (hi)i∈N be orthonormal bases for a separable Hilbert space H. Let ( fi)i∈Nbe a sequence such that for every j ∈ N,

∑i∈N |⟨ fi, e j⟩|2 < ∞ and

ωfj =

∑i∈N⟨ fi, e j⟩hi.

The sequence (ω fj ) j∈N is called the R-dual sequence of ( fi)i∈N with respect to (e j) j∈N

and (hi)i∈N.But it is still an open problem whether the duality principle in Gabor analysis actuallycan be derived from the theory of the R-dual. Lots of scholars have done much research∗ speaker

112

A. Khosravi and F. Takhteh

in this area, see [2],[3], [5], [6].In this paper, for every Bessel sequence and every Riesz basis, we consider an anti-linear operator on H. We give important characterizations of frames, Riesz bases,Riesz sequences, and Besselian frames in terms of this anti-linear map, then weconsider the concept of R-duality with respect to Riesz bases and we apply our resultsto characterizing of frames, Riesz bases, Riesz sequences in terms of their R-dualsequences with respect to Riesz bases.In the rest of this section, we briefly recall the basic definitions of frame theory. Insection 2, we define the anti-linear map M and we characterize frames, Riesz bases,Riesz sequences, and Besselian frames in terms of M. In section 3, we apply ourresults in section 2, and we give important characterizations of frames, Riesz bases,Riesz sequences.Throughout this paper H denotes a separable Hilbert space with inner product ⟨., .⟩ andI denotes a subset of Z.A sequence ( fi)i∈I in H is a frame if there exist constants A, B > 0 such that for everyf ∈ H,

A∥ f ∥2 ≤∑i∈I|⟨ f , fi⟩|2 ≤ B∥ f ∥2.

The numbers A, B are called frame bounds. A frame is tight if A = B. A sequence( fi)i∈I is called a Bessel sequence if the right hand side inequality is fulfilled.A sequence ( fi)i∈I in H is a Riesz sequence if there exist A, B > 0 such that for everysequence of scalars (ci)i∈I ∈ ℓ2(I)

A∑i∈I|ci|2 ≤ ∥

∑i∈I

ci fi∥2 ≤ B∑i∈I|ci|2.

A Riesz sequence ( fi)i∈I is a Riesz basis for H if span( fi)i∈I = H.We say that a frame (xn)n∈I, for H is a Besselian frame, if whenever

∑n∈I anxn

converges, then (an)n∈I ∈ ℓ2.

2. characterizations of Bessel sequences

In this section, we define an anti-linear map as follows:Let ( fi)i∈I be a Bessel sequence in H with Bessel bound A and (hi)i∈I be a Riesz basis.Define the anti-linear operator M : H → H by

M( f ) =∑i∈I⟨ fi, f ⟩hi ( f ∈ H) (2.1)

M is a well-defined bounded anti-linear operator on H. Its adjoint M∗ is an anti-linearoperator and

M∗ f =∑i∈I⟨hi, f ⟩ fi ( f ∈ H).

In the following proposition, we give a characterization of Bessel sequences.

113


Proposition 2.1. ( fi)i∈I is a Bessel sequence if and only if for every f ∈ H,∑i∈I |⟨ fi, f ⟩|2 < ∞.

Proof. The necessary condition is obvious. We prove the sufficient condition. Sincefor every f ∈ H,

∑i∈I |⟨ fi, f ⟩|2 < ∞ the operator U : H → ℓ2 defined by

U( f ) = (⟨ fi, f ⟩)i∈I is well-defined and linear. Easily, we can see the graph of U isclosed, then by Closed Graph Theorem U is a bounded operator. Hence, ( fi)i∈I is aBessel sequence.

In the following Theorem, we give characterization of frames, Riesz bases, Rieszsequences in terms of M defined in (2.1).

Theorem 2.2. Let ( fi)i∈I be a Bessel sequence with Bessel bound A, (hi)i∈I and (e j) j∈Ibe Riesz bases in H and M be defined as (2.1). Then the following statements hold:1. ( fi)i∈I is a frame for H if and only if M is bounded below.2. ( fi)i∈I is a Riesz basis in H if and only if M is an invertible anti-linear map .

Moreover, for every f ∈ H

M−1( f ) =∑i∈I⟨hi, f ⟩ fi.

3. ( fi)i∈I is a Riesz sequence in H if and only if M is a surjective anti-linear map.4. ( fi)i∈I is a frame sequence for H if and only if R(M) is closed.5. ( fi)i∈I is a Besselian frame in H if and only if M∗ is a surjective Fredholm anti-

linear map on H.

Proposition 2.3. Let ( fi)i∈I be a Bessel sequence in H and Let (hi)i∈I be an orthonor-mal basis for H. Let M be defined as (2.1). Then the following statements hold:1. ( fi)i∈I is an orthonormal basis for H if and only if M is a unitary anti-linear

operator.2. ( fi)i∈I is a Parseval frame for H if and only if M is an isometry and anti-linear

operator.

Remark 2.4. Let ( fi)i∈I be a Bessel sequence in H and Let (hi)i∈I be a Riesz basis inH. Let M be defined as (2.1). If M is compact anti-linear operator, then ( fi)i∈I is arelatively compact in H.

It is easy to see that M((hi)i∈I) = ( fi)i∈I. Since (hi)i∈I is a bounded sequence forH, by the compactness of M, ( fi)i∈I is relatively compact in H.

3. R-duality with respect to Riesz bases

In this section, first we define the concept of R-duality with respect to Riesz bases.

Definition 3.1. Let (e j) j∈I and (hi)i∈I be Riesz bases for H. Let ( fi)i∈I be a Besselsequence in H and

ωfj =

∑i∈I⟨ fi, e j⟩hi.

114

A. Khosravi and F. Takhteh

Then (ω fj ) j∈I is called R-dual sequence of ( fi)i∈I with respect to (e j) j∈I and (hi)i∈I.

Remark 3.2. Let ( fi)i∈I be a Bessel sequence and (e j) j∈I and (hi)i∈I be Riesz basesin H. Then the R-dual sequence of ( fi)i∈I with respect to (e j) j∈I and (hi)i∈I is(M(e j)) j∈I = (ω f

j ) j∈I, where M is defined as (2.1).

In the rest of this section, we apply our results in section 2 and we characterizeframes and Riesz bases in terms of their R-dual sequences with respect to Riesz bases

Theorem 3.3. Let (e j) j∈I and (hi)i∈I be Riesz bases in H, ( fi)i∈I be a Bessel sequenceand (ω f

j ) j∈I be the R-dual sequence of ( fi)i∈I with respect to (e j) j∈I and (hi)i∈I. Thenthe following statements are equivalent:1. ( fi)i∈I is a Riesz basis in H.2. (ω f

j ) j∈I is a Riesz basis in H.

Proof. Since ( fi)i∈I is a Bessel sequence if and only if (ω fj ) j∈I is a Bessel sequence.

Let M be defined as (2.1). For every g ∈ H, we have

M1(g) =∑j∈I⟨ω f

j , g⟩e j =∑j∈I⟨M(e j), g⟩e j =

∑j∈I⟨M∗(g), e j⟩e j = M∗(g)

Therefore M1 = M∗. By Theorem 2.2, ( fi)i∈I is a Riesz basis if and only if M isinvertible. Similarly (ω f

j ) j∈I is a Riesz basis if and only if M1 is invertible. SinceM∗ = M1, then M is invertible if and only if M1 is invertible. Therefore ( fi)i∈I is aRiesz basis if and only if (ω f

j ) j∈I is a Riesz basis.

Theorem 3.4. Let (e j) j∈I and (hi)i∈I be Riesz bases in H, ( fi)i∈I be Bessel sequenceand (ω f

j ) j∈I be the R-dual sequence of ( fi)i∈I with respect to (e j) j∈I and (hi)i∈I. Then

( fi)i∈I is a frame in H if and only if (ω fj ) j∈I is a Riesz sequence in H.

Proof. A proof similar to the proof of Theorem 3.3, and using the fact M is boundedbelow if and only if M∗ = M1 is surjective, proves the claim.

[1] P. Balazs: Basic definition and properties of Bessel multipliers. Journal of MathematicalAnalysis and Applications., 325 (2007), 571–585.

[2] P. Casazza, G. Kutyniok, and M.C. Lammers, Duality principles in frame theory, Journal ofFourier Analysis and Applications, 10 (2004), 383–408.

[3] P. Casazza, G. Kutyniok, andM.C. Lammers, Duality principle, localization of frames, and Gabortheory. Optics and photonics, International society for optics and photonics, 2005.

[4] O. Christensen: Frames and bases. An introductory course. Birkhäuser, Boston, 2008.[5] O. Christensen, HO Kim, and RY Kim: On the duality principle by Casazza, Kutyniok, and

Lammers, Journal of Fourier Analysis and Applications, 17 (2011), 640–655.[6] Z. Chuang, j. Zhao , On equivalent conditions of two sequences to be R-dual, Journal of

Inequalities and Applications 1 (2015), 1–8.

115


[7] D. Han, D. R. Larson: Frames, bases and group representations. American Mathematical Society,697 2000.

[8] A.Ron, Z. Shen, Weyl-Heisenberg frames and Riesz bases in L2(Rd),Duke Mathematical Journal 89 (1997), 237–282.

[9] J. Wexler and S. Raz, Discrete Gabor expansions, Signal Processing 21 (1990), 207–227.

A. Khosravi,Faculty of Mathematical Sciences and Computer,University of Kharazmi,City Tehran, Irane-mail: [email protected]

F. Takhteh,Faculty of Mathematical Sciences and Computer,University of Kharazmi,City Tehran, Irane-mail: [email protected]

116



January 20–21, 2016

On the Converse of a Theorem Due to B. Forrest, E. Kaniuth, A.T.-M. Lau and N. Spronk

J. Laali and M. Fozouni∗

AbstractIt was shown by Forrest et al. (2003) that if G is an amenable locally compact group, H is a closedsubgroup of G and 1 < p < ∞, then Ip(H) = u ∈ Ap(G) : u(x) = 0 (x ∈ H) has a bounded approximateidentity. We show that the converse of this assertion is also holds, that is, if H is a closed proper subgroupof G such that Ip(H) has a bounded approximate identity, then G is amenable.

2010 Mathematics subject classification: Primary 43A07, Secondary 43A15.Keywords and phrases: Amenable group, Figà-Talamanca-Herz algebra, Character space.

1. Introduction

Let A be a Banach algebra and ∆(A) be the character space of A, that is, the space ofall non-zero homomorphisms from A into C.

Let eα be a bounded net in Banach algebra A. The net eα is called,1. an bounded approximate identity (b.a.i) if for each a ∈ A, ∥aeα − a∥ → 0,2. a bounded weak approximate identity (b.w.a.i) if for each ϕ ∈ ∆(A), |ϕ(eα)−1| →

0.The notion of a weak approximate identity introduced and studied in [7]. These type ofapproximate identities have some interesting applications, for example; see [6, 10, 15].In the past decades, B. E. Forrest studied the relations between the amenability of agroup G and closed ideals of Ap(G) with a b.a.i; see [2–5]. If G is an amenable locallycompact group, H is a closed subgroup of G and 1 < p < ∞, then the closed idealIp(H) = u ∈ Ap(G) : u(x) = 0 (x ∈ H) has a b.a.i; see [5, Corollary 4.2]. Here Ap(G)denotes the Figà-Talamanca-Herz algebra.

In this paper, we give the converse of the above result due to B. Forrest, E. Kaniuth,A. T. Lau and N. Spronk.

2. Main Result

The following theorem is due to Kaniuth and Ülger in the case p = 2. In generalthe proof is similar, therefore we omit it; see [12, Theorem 5.1].∗ speaker

117

J. Laali andM. Fozouni

Theorem 2.1. Let G be a locally compact group and 1 < p < ∞. Then Ap(G) has ab.w.a.i if and only if G is amenable.

The following theorem is a key tool in the sequel.

Theorem 2.2. Let A be a Banach algebra, I be a closed two-sided ideal of A whichhas a b.w.a.i and the quotient Banach algebra A/I has a b.l.a.i. Then A has a b.w.a.i.

Proof. Let eα be a b.w.a.i for I and fδ + I be a b.l.a.i for A/I. We can assume that fδ is bounded. Indeed, since fδ+ I is bounded, there exists a positive integer K with∥ fδ + I∥ < K for each δ. So, there exists yδ ∈ I such that ∥ fδ + I∥ < ∥ fδ + yδ∥ < K. Putf′

δ = fδ + yδ. Clearly, f ′δ + I is a b.l.a.i for A/I which f ′δ is bounded.Now, consider the bounded net eα + fδ − eα fδ(α,δ). For each ϕ ∈ ∆(A) we have

ϕ(eα + fδ − eα fδ) = ϕ(eα) + ϕ( fδ)(1 − ϕ(eα))(α,δ)−−−→ 1.

Therefore, A has a b.w.a.i.

The following result improves [2, Theorem 3.9].

Theorem 2.3. Let G be a locally compact group. Then the following assertions areequivalent.

1. G is an amenable group.2. ker(ϕ) has a b.w.a.i for each ϕ ∈ ∆(Ap(G)).3. Ip(H) has a b.w.a.i for some closed amenable subgroup H of G.

Proof. (1) ⇒ (2): Let G be an amenable group. Then Ap(G) has a b.a.i by Leptin-Herz’s Theorem. Now, the result follows from [11, Corollary 2.3].

(2)⇒ (3): Just take H = e, because we know that Ip(e) = ker(ϕe).(3)⇒ (1): Suppose that Ip(H) for a closed amenable subgroup H of G has a b.w.a.i.

By [14, Lemma 3.19] we know that Ap(H) is isometrically isomorphic to Ap(G)/Ip(H).But Ap(H) has a b.a.i, since H is an amenable group. Therefore, Ap(G)/Ip(H) also hasa b.a.i. Now, the result follows from Theorems 2.2 and 2.1.

Now, we give the following result which improves [5, Corollary 1.6] and Theorem2.3.

Theorem 2.4. Let G be a locally compact group and 1 < p < ∞. Then the followingare equivalent.

1. G is an amenable group.2. Ip(H) has a b.w.a.i for some proper closed subgroup H of G.

Proof. In view of [5, Corollary 4.2], only (2)⇒ (1) needs proof.Let H be a proper closed subgroup of G such that Ip(H) has a b.w.a.i. We will show

that H is an amenable group and this completes the proof by Theorem 2.3.

118

On the Converse of a Theorem due to B. Forrest et al.

Since H is a proper subgroup, there exists x ∈ G \ H. On the other hand, themapping Ip(H) → Ip(xH) defined by u → Lxu is an isometric isomorphism, becausefor each t ∈ G and f ∈ Ap(G), we have,

Lt f ∈ Ap(G), ∥Lt f ∥Ap(G) = ∥ f ∥Ap(G).

Therefore, Ip(xH) has a b.w.a.i which we denote it by (uα). For each α, let vα bethe restriction of uα to H. Using [8, Theorem 1a], we conclude that (vα) is a boundednet in Ap(H).

Let ν ∈ Ap(H) ∩ Cc(H) and K = supp ν ⊆ H. Then there exists a neighborhoodV of K in G such that V ∩ xH = ∅, because K ∩ xH = ∅ (otherwise we conclude thatx is in H) and G is completely regular by [9, Theorem 8.4] and hence it is a regulartopological space. Indeed, for each y ∈ xH, let Vy be a neighborhood of K such thaty < Vy. So, V = ∩y∈xHVy satisfies V ∩ xH = ∅.

By [1, Proposition 1, pp.34] there is u ∈ Ap(G) such that u(x) = 1 for each x ∈ Kand supp u ⊆ V , and by [8, Theorem 1b], there exists a v ∈ Ap(G) such that v|H = ν.Now, put w = vu. Since V ∩ xH = ∅ and supp u ⊆ V , we have w ∈ Ip(xH), and sinceu(x) = 1 for each x ∈ K and K = supp ν, we have w|H = ν.

Now, for each x ∈ H, we have

limα|ϕx(vαν) − ϕx(ν)| = lim

α|vα(x)ν(x) − ν(x)|

= limα|uα(x)w(x) − w(x)| = 0.

Therefore, (vα) is a b.w.a.i for Ap(H), since by [1, Corollary 7, pp. 38], Ap(H)∩Cc(H)is dense in Ap(H). Hence, by Theorem 2.1, H is amenable.

As an application of the above theorem, we give the following corollary which isthe converse of [5, Corollary 4.2].

Corollary 2.5. Let G be a locally compact group, 1 < p < ∞ and H be a proper closedsubgroup of G. If Ip(H) has a b.a.i, then G is amenable.

[1] A. Derighetti, Convolution Operators on Groups, Springer-Verlag, 2011.[2] B. E. Forrest, Amenability and bounded approximate identities in ideals of A(G), Illinois J.

Math., 34 (1990) 1–25.[3] B. E. Forrest, Amenability and ideals in A(G), J. Austral. Math. Soc. (Series A) 53 (1992)

143–155.[4] B. E. Forrest, Amenability and the structure of the algebras Ap(G), Trans. Amer. Math. Soc.,

343 No. 1 (1994) 233–243.[5] B. Forrest, E. Kaniuth, A.T. Lau and N. Spronk, Ideals with bounded approximate identities in

Fourier algebras, J. Funct. Anal. 203 (2003) 286–304.[6] B. Forrest, M. Skantharajah, A note on a type of approximate identity in the Fourier algebra,

Proc. Amer. Math. Soc. 120, No. 2 (1994) 651–652.[7] C. A. Jones, C. D. Lahr, Weak and norm approximate identities, Pacific J. Math., 72(1) (1977)

99–104.

119

J. Laali andM. Fozouni

[8] C. Herz, Harmonic synthesis for subgroups, Ann. Inst. Fourier (Grenoble) Vol XXIII, 3 (1973)91–123.

[9] E. Hewitt, K. Ross, Abstract Harmonic Analysis I, Die Grundlehren der MathematischenWissenschaften, 115. Springer-Verlag, Berlin, Second edition 1979.

[10] Z. Kamali, M. L. Bami, Bochner-Schoenberg-Eberlein property for abstract Segal algebras, Proc.Japan Acad. Ser A. 89,(2013), 107–110.

[11] E. Kaniuth, A. T. Lau and J. Pym, On ϕ-amenability of Banach algebras, Math. Proc. Camb. Phil.Soc. (2008), 144, 85–96.

[12] E. Kaniuth, A. Ülger, The Bochner-Schoenberg-Eberlian property for commutative Banachalgebras, especially Fourier-Stieltjes algebras, Trans. Amer. Math, Soc. 362, 2010, 4331–4356.

[13] J. Laali, M. Fozouni, Closed ideals with bounded ∆-weak approximate identities in some certainBanach algebras, MiskolcMath. Notes, to appear.

[14] M. Sangani Monfared, Extensions and isomorphisms for the generalized Fourier algebras of alocally compact group, J. Funt. Anal. 198 (2003) 413–444.

[15] S. Takahasi, O. Hatori, Commutative Banach algebras which satisfy a Bochner-Schoenberg-Eberlein type-theorem, Proc. Amer. Math. Soc. 110 (1990), 1, 149–158.

J. Laali,Department of Mathematics,Kharazmi University,Tehran, Irane-mail: [email protected]

M. Fozouni,Department of Mathematics,Gonbad Kavous University,Gonbad Kavous, Irane-mail: [email protected]

120



January 20–21, 2016

∗-Fusion frames in Hilbert modules over locally C∗-algebras

T. Lal Shateri

AbstractThe main purpose of this paper is to introduce the notion of ∗-fusion frames in Hilbert modules overlocally C∗-algebras to study some properties about these frames.

2010 Mathematics subject classification: Primary 42C15 Secondary 06D22.Keywords and phrases: Locally C∗-algebra, Hilbert module, ∗-fusion frame.

1. IntroductionFrames for Hilbert spaces were introduced in 1952 by Duffin and Schaeffer [5] tostudy some problems in nonharmonic Fourier series. Then Daubecheies, Grassmanand Mayer [4] reintroduced and developed them. Various generalizations of framese.g. frames of subspaces, g-frames were developed [3, 9, 11]. Frank and Larson [6]presented a general approach to the frame theory in Hilbert C∗-modules. A. Khosraviand B. Khosravi [8] generalized the concept of fusion frames and g-frames to HilbertC∗-modules. A. Alijani and M.A. Dehghan [1] introduced the ∗-frames and studied theproperties of them in Hilbert C∗-modules. Finally, M. Azhini and N. Haddadzadeh [2]generalized the theory of fusion frames to Hilbert modules over locally C∗-algebras.

In this paper, we introduce ∗-fusion frames for Hilbert modules over locally C∗-algebras and give some results about them.

First, locally C∗-algebras and Hilbert modules over them are defined. Recall that aC∗-seminorm on a topological ∗-algebraA is a seminorm p such that p(ab) ≤ p(a)p(b)and p(aa∗) = (p(a))2 for all a, b ∈ A.

Definition 1.1. A locally C∗-algebra is a Hausdorff complete complex topological ∗-algebraA whose topology is determined by its continuous C∗-seminorms.

In the sense that a net aαα∈I converges to 0 if and only if the net p(aα)α∈Iconverges to 0 for all continuous C∗seminorm p onA.Note that, each C∗-algebra is a locally C∗-algebra.

The set of all continuous C∗-seminorms onA is denoted by S (A).LetA be a unital locally C∗-algebra with unit 1A and a ∈ A. a is called positive if

a∗ = a and sp(a) = λ ∈ C : λ1A − a is not invertible ⊆ R+. The set of all positiveelements ofA denotes byA+. If a, b ∈ A, then a ≤ b means that b − a ∈ A+.

121

T. Lal Shateri

Proposition 1.2. ([7]) LetA be a unital locally C∗-algebra with unit 1A. Then for anyp ∈ S (A) and a, b ∈ A, the following holds:(1) p(a) = p(a∗)(2) p(1A) = 1(3) If a, b ∈ A+ and a ≤ b, then p(a) ≤ p(b)(4) If 1A ≤ b, then b is invertible and b−1 ≤ 1A(5) If a, b ∈ A+ are invertible and 0 ≤ a ≤ b, then 0 ≤ b−1 ≤ a−1

(6) If a, b are self adjoint, a ≤ b and c ∈ A, then c∗ac ≤ c∗bc(7) If a, b ∈ A+ and a2 ≤ b2, then 0 ≤ a ≤ b.

Now, we recall some definitions and basic properties of Hilbert modules overlocally C∗-algebras from [12].

Definitions 1.3. A pre-Hilbert module over locally C∗-algebra A is a complex vectorspace X which is also a left A-module equipped with an A-valued inner product< ., . >: X×X → A which is C-linear andA-linear in its first variable and satisfy thefollowing conditions:(i) < x, x >≥ 0,(ii) < x, x >= 0 iff x = 0,(iii) < x, y >∗=< y, x >,for all x, y ∈ X.

A pre-Hilbert A-module X is called Hilbert A-module if X is complete withrespect to the topology determined by the family of seminorms

pX(x) =√

p(< x, x >) (x ∈ X, p ∈ S (A)).

If A is a locally C∗-algebra, then it is a Hilbert A-module with respect to the innerproduct < a, b >= ab∗ (a, b ∈ A).

Lemma 1.4. [12, Lemma 2.1] For every p ∈ S (A) and for all x, y ∈ X, the Cauchy-Bunyakovskii inequality holds

p(< x, y >)2 ≤ p(< x, x >)p(< y, y >).

Example 1.5. Let l2(A) be the set of all sequences ann∈N of elements of a locally C∗-algebra A such that the series

∑∞n=1 ana∗n is convergent in A. Then l2(A) is a Hilbert

A-module with respect to the pointwise operations and inner product defined by

< ann∈N, bnn∈N >=∞∑

n=1

anb∗n.

Definition 1.6. Let M be a closed submodule of a HilbertA-module X. Define

M⊥ = y ∈ X : < x, y >= 0, for all x ∈ M.Then M⊥ is a closed submodule of X.A closed submodule M in a Hilbert A-module X is called orthogonally comple-

mented if X = M ⊕ M⊥.

122

∗-Fusion frames in Hilbert modules over locally C∗-algebras

2. ∗-fusion frames in Hilbert modules over locally C∗-algebras

In this section, we assume that A is a unital locally C∗-algebra and X is a HilbertA-module. We introduce ∗-fusion frames in Hilbert modules over locally C∗-algebras,and then we give some results about them.

Definition 2.1. Let vi ∈ A : i ∈ I be a sequence of weights in A, that is each vi isa positive invertible element from the center of A, and let Mi : i ∈ I be a sequenceof orthogonally complemented submodules of X. Then (Mi, vi) : i ∈ I is called a∗-fusion frame if there are two strictly nonzero elements C,D ∈ A such that

C < x, x > C∗ ≤∑i∈I

v2i < PMi (x), PMi (x) >≤ D < x, x > D∗, (x ∈ X), (1)

where PMi is the orthogonal projection of X onto Mi.

We call C and D the lower and upper bounds of the ∗-fusion frame. SinceA is not apartial ordered set, lower and upper ∗-frame bounds may not have order. If C = D = λ,the family (Mi, vi) : i ∈ I is called a λ-tight ∗-fusion frame and if C = D = 1A, itis called a Parseval ∗-fusion frame. If in (1), we only have the upper bound, then(Mi, vi) : i ∈ I is called a ∗-Bessel fusion sequence with ∗-Bessel bound D. Now, wegive some results about ∗-fusion frames.

Proposition 2.2. Let X be a Hilbert A-module on a commutative C∗-algebra A andlet vi : i ∈ I be a family of weights in A. Let for each i ∈ I, Mi be an orthogonallycomplemented submodule of X and let xi j : j ∈ Ji be a frame for Mi with positivebounds Ci and Di. Suppose that C2

i ≥ 1A for each i and supi∈I ∥Di∥ < ∞, then thefollowing conditions are equivalent.(i) vixi j : i ∈ I; j ∈ Ji is a ∗-frame for X.(ii) (Mi, vi) : i ∈ I is a ∗-fusion frame for X.

Now, We generalize [6, Theorem 4.1] to ∗-Bessel fusion frames.

Theorem 2.3. Let (Mi, vi) : i ∈ I be a ∗-Bessel fusion frame for a Hilbert A-moduleX with ∗-Bessel bound D. Then, the corresponding frame transform θ : X → l2(X)defined by θ(x) = (viPMi(x))i∈I for x ∈ X, is an isomorphic imbedding with closedrange, and its adjoint operator θ∗ : l2(X) → X defined as θ∗(y) =

∑i∈I viPMi (yi) for

each y = (yi)i∈I ∈ l2(X), is bounded.

The following propositions follow from the definitions.

Proposition 2.4. Let X be a Hilbert A-module and let (Mi, vi) : i ∈ I be a ∗-fusionframe for X with frame bounds C and D. If M is an orthogonally complementedsubmodule of X. Then (Mi ∩ M, vi) : i ∈ I is a ∗-fusion frame for M with framebounds C and D.

Proposition 2.5. Let (Mi, vi) : i ∈ I be a Parseval ∗-fusion frame for a Hilbert A-module X. Then, the corresponding frame transform θ preserves the inner product.

123

T. Lal Shateri

Definition 2.6. Let (Mi, vi) : i ∈ I be a ∗-fusion frame for a Hilbert A-module X.Then the fusion frame operator S for (Mi, vi) : i ∈ I is defined by

S (x) = θ∗θ(x) =∑i∈I

v2i PMi (x), (x ∈ X).

Our next result is a generalization of [8, Theorem 2.11] for ∗-fusion frames withinvertible ∗-fusion frame bounds.

Theorem 2.7. (Reconstruction formula) Let (Mi, vi) : i ∈ I be a ∗-fusion frame fora Hilbert A-module X with ∗-fusion frame operator S and invertible ∗-fusion framebounds C and D. Then, S is a positive, self-adjoint and invertible operator on X suchthat for all x ∈ X,

x =∑i∈I

v2i S −1PMi (x).

[1] A. Alijani and M.A. Dehghan, ∗-Frames in Hilbert C∗-modules, U.P.B. Sci. Bull. Series A 73(4)(2011) 89-106.

[2] M. Azhini and N. Haddadzadeh, Fusion frames in Hilbert modules over pro-C∗-algebras, Int. J.Industrial Mathematics 5 (2013) Article ID: IJIM-00211.

[3] P. G. Casazza and G. Kutyniok, Frames of subspaces, in Wavelets, Frames and Operator Theory,Contemp. Math. 345 (2004) 87-113.

[4] I. Daubechies, A. Grassman and Y. Meyer, Painless nonothogonal expanisions, J. Math. Phys. 27(1986) 1271-1283.

[5] R.J. Duffin and A.C. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc.72 (1952) 341-366.

[6] M. Frank and D. R. Larson, Frames in Hilbert C∗-modules and C∗-algebras, J. Operator Theory48 (2002) 273-314.

[7] A. Inoue, Locally C*-algebras, Mem. Fac. Sci. Kyushu Univ. Ser. A 25 (1971) 197-235.[8] A. Khosravi and B. Khosravi, Fusion frames and g-frames in Hilbert C∗-Modules, Int. J. Wavelet,

Multiresolution and Information Processing 6(3) (2008) 433-446.[9] A. Nejati and A. Rahimi, Generalized frames in Hilbert spaces, Bull. Iranian Math. Soc. 35 (1)

(2009) 97-109.[10] N.C. Phillips, Inverse limits of C*-algebras, J. Operator Theory 19 (1988) 159-195.[11] W. Sun, G-frames and g-Riesz bases, J. Math. Anal. Appl. 322 (2006) 437-452.[12] Yu. I. Zhuraev and F. Sharipov, Hilbert modules over locally C*-algebra, Preprint, posted on

Arxiv Org. Math. OA/0011053 V3 (2001).

T. Lal Shateri,Department of Mathematics and Computer Sceinces,Hakim Sabzevari University,City Sabzevar, Irane-mail: [email protected]

124



January 20–21, 2016

Dual Banach algebras; a type of φ-contractibility

A. Mahmoodi

Abstract

Let φ be a w∗-continuous homomorphism from a dual Banach algebra to C. The notion of φ-Connesamenability is introduced. It is shown that φ-Connes amenability is equivalent to φ-contractibility.For a general Banach algebra A, φ-contractibility of the enveloping dual Banach algebra WAP(A∗)∗is discussed. The results are applied for some certain (dual) Banach algebras.

2010 Mathematics subject classification: Primary: 22D15, 43A10; Secondary: 43A20, 46H25.Keywords and phrases: dual Banach algebra, φ-amenability, φ-contractibility, φ-injectivity.

1. Introduction

Amenability for Banach algebras as introduced by B. E. Johnson [7], has proved tobe an important and fertile notion. A generalization of amenability which depends onhomomorphisms was introduced and studied by E. Kaniuth, A. T. Lau and J. Pym in[8,9]. This concept was also studied independently, by M. S. Monfared in [14]. LetA be a Banach algebra and φ be a homomorphism from A onto C. We say A is φ-amenable if there exists a bounded linear functional m onA∗ satisfying m(φ) = 1 andm( f . a) = φ(a)m( f ), for all a ∈ A and f ∈ A∗. We write ∆(A) for the set of allhomomorphism from A onto C. Recently, the notion of φ-contractibility of Banachalgebras was introduced by Z. Hu, M. S. Monfared and T. Traynor [6]. A Banachalgebra A is φ-contractible if every derivation D : A −→ E is inner, where E is aBanach A-bimodule such that x . a = φ(a)x for all a ∈ A and x ∈ E. In fact, sayingthat A is φ-contractible is equivalent to saying that there exists u ∈ A for whichφ(u) = 1 and au = φ(a)u, for all a ∈ A.

Let A be a Banach algebra. A Banach A-bimodule E is dual if there is aclosed submodule E∗ of E∗ such that E = (E∗)∗. A Banach algebra is dual if itis dual as a Banach A-bimodule. Let A be a dual Banach algebra. Then a dualBanach A-bimodule E is normal if the module actions of A on E are w∗-continuous.Systematically, the concept of Connes amenability introduced by V. Runde [15],although it had been studied previously under different names (see for instance [5]). Adual Banach algebraA is Connes amenable if every w∗-continuous derivation fromAinto a normal, dual BanachA-bimodule is inner. One may see [2,3,10,11,13,15,16,17]

125

A. Mahmoodi

for further information on this subject and its generalizations. For a dual Banachalgebra A, ∆w∗(A) will denote the set of all w∗-continuous homomorphism from Aonto C. Let A be a dual Banach algebra and let E be a Banach A-bimodule. Thenσwc(E) stands for the set of all elements x ∈ E such that the maps

A −→ E , a 7−→

a . xx . a ,

are w∗-weak continuous.In this note, for a dual Banach algebra A and for φ ∈ ∆w∗(A), we show that the

following three statements are equivalent (Theorems 2.2 and 3.6 below). (i) A isφ-contractible; (ii) A is φ-Connes amenable; and (iii) A is φ-injective. We brieflysummarize the other results in this paper. LetA be a Banach algebra, φ ∈ ∆(A), and φbe the unique extension of φ to a w∗-continuous homomorphism from the envelopingdual Banach algebra WAP(A∗)∗ onto C. Theorem 4.5 characterizes φ-contractibilityof WAP(A∗)∗ in terms of continuous representations from A on reflexive Banachspaces. We investigate the relation between φ-amenability/contractibility of A andφ-contractibility of WAP(A∗)∗ (Theorems 4.1 and 4.2). Then, in the light of theresults, we discuss φ-amenability/contractibility of some concrete Banach algebras.We show that WAP(ℓ1(Z+)∗)∗ is φ-contractible for some suitable φ, however it is notcontractible or even amenable (Example 4.3). For a locally compact group G, we lookat φ-contractibility of WAP(VN(G))∗ (Example 4.4).

Let N∧ and N∨ be the semigroup N with products m ∧ n = minm, n, andm ∨ n = maxm, n, respectively. Theorem 4.6 shows that WAP(ℓ1(N∧)∗)∗ is not φ-contractible, where φ is the augmentation character. As a consequence, we observe thatWAP(ℓ1(N∨)∗)∗ is not 0-contractible (Corollary 4.7). Example 4.8 says that ℓ1(N∧) isnot φ-amenable. It is known that ℓ1(N∨) is φ-amenable, for every φ ∈ ∆(ℓ1(N∨)) [4,Corollary 2.2]. We show that ℓ1(N∨) is not 0-amenable (Examples 4.9).

2. φ-Contractibility

We start with the following observation. Suppose that A is a dual Banach algebraand φ is a homomorphism from A onto C. Then φ is w∗-continuous if and only ifφ ∈ σwc(A∗).

Definition 2.1. Suppose that A is a dual Banach algebra and φ ∈ ∆w∗(A). We callA φ-Connes amenable if there exists a bounded linear functional m on σwc(A∗)satisfying m(φ) = 1 and m( f . a) = φ(a)m( f ) for all a ∈ A and f ∈ σwc(A∗).

The following is the key result of the current section.

Theorem 2.2. Suppose that A is a dual Banach algebra and φ ∈ ∆w∗(A). Then thefollowing are equivalent

(i)A is φ-contractible;(ii)A is φ-Connes amenable.

126

Dual Banach algebras; φ-contractibility

We have the following hereditary property.

Theorem 2.3. Suppose that A is a Banach algebra, B is a dual Banach algebra, θ :A −→ B is a continuous homomorphism with w∗-dense range, and that φ ∈ ∆w∗(B).

(i) IfA is φ θ-amenable, then B is φ-contractible.(ii) IfA is a dual Banach algebra and θ is w∗-continuous, then φ θ-contractibility

ofA implies φ-contractibility of B.

LetA be an Arens regular Banach algebra which is an ideal inA∗∗. It is immediatethat A∗∗ is a dual Banach algebra [16]. Let φ ∈ ∆(A). Then φ, the extension of φ toA∗∗, belongs to ∆w∗(A∗∗).

Theorem 2.4. Let A be an Arens regular Banach algebra which is an ideal in A∗∗,and let φ ∈ ∆(A). Then the following are equivalent:

(i)A is φ-amenable.(ii)A∗∗ is φ-contractible.

3. φ-Injectivity

Let S be a subset of an algebraH . We use Sc to denote the commutant of S inH ,i.e., Sc = h ∈ H : hs = sh, s ∈ S. It is obvious that Sc is a closed subalgebra ofH . Let be a Banach algebra, and let S be a closed subalgebra of . It is standard thatL(E) = (E∗⊗E)∗ is a dual Banach algebra, whenever E is a reflexive Banach space[15].

Let A and B be Banach algebras, and let θ : A −→ B be a homomorphism. Forφ ∈ ∆(A), we define

θ(A)φ = b ∈ B : θ(a)b = φ(a)b (a ∈ A) .

Obviously θ(A)φ is a closed subalgebra of B.

Definition 3.1. LetA andB be Banach algebras, let θ : A −→ B be a homomorphism,and let φ ∈ ∆(A). A φ-quasi expectation Q : B −→ θ(A)φ is a projection from B ontoθ(A)φ satisfying Q(cbd) = cQ(b)d, for c, d ∈ θ(A)c and b ∈ B.

Definition 3.2. Let A be a dual Banach algebra, and let φ ∈ ∆w∗(A). We say thatA is φ-injective if whenever π : A −→ L(E) is a w∗-continuous representation on areflexive Banach space E, then there is a φ-quasi expectation Q : L(E) −→ π(A)φ.

Let A be a dual Banach algebra. It is known that its unitization A♯ = A ⊕ Ce is adual Banach algebra as well. Let φ ∈ ∆w∗(A) and let φ♯ be its unique extension toA♯.It is obvious that φ♯ ∈ ∆w∗(A♯).

Theorem 3.3. Suppose that A is a dual Banach algebra and that φ ∈ ∆w∗(A). ThenA is φ-injective if and only ifA♯ is φ♯-injective.

127

A. Mahmoodi

Theorem 3.4. Suppose that A is a Banach algebra, B is a dual Banach algebra, andθ : A −→ B is a continuous homomorphism. Suppose that one of the following holds:

(i) φ ∈ ∆(A) andA is φ-amenable;(ii) A is a dual Banach algebra, θ is w∗-continuous, φ ∈ ∆w∗(A), and A is φ-

contractible.Then there is a φ-quasi expectation Q : B −→ θ(A)φ.

Theorem 3.5. Suppose that A is a dual Banach algebra and φ ∈ ∆w∗(A). Then A isφ-contractible if and only ifA♯ is φ♯-contractible.

Theorem 3.6. Suppose that A is a dual Banach algebra and φ ∈ ∆w∗(A). Then A isφ-injective if and only ifA is φ-contractible.

4. Application to W AP(A∗)∗ and examples

Suppose that A is a Banach algebra and that E is a Banach A-bimodule. Anelement x ∈ E is weakly almost periodic if the maps

A −→ E , a 7−→

a . xx . a ,

are weakly compact. The set of all weakly almost periodic elements in E is denotedby WAP(E). For a Banach algebra A, there is a well-defined product on WAP(A∗)∗turning it into a dual Banach algebra with a universal property; for every dual Banachalgebra B and every homomorphism θ : A −→ B there is a unique w∗-continuoushomomorphism θ : WAP(A∗)∗ −→ B such that θ ı = θ, where ı : A −→ WAP(A∗)∗is the canonical map [17, Theorem 4.10]. For φ ∈ ∆(A), we consider the uniqueelement φ ∈ ∆w∗(WAP(A∗)∗) such that φ = φ ı.Theorem 4.1. LetA be a Banach algebra, and let φ ∈ ∆(A). IfA is φ-amenable, thenWAP(A∗)∗ is φ-contractible.

Theorem 4.2. Let A be a dual Banach algebra, and let φ ∈ ∆w∗(A). Then A is φ-contractible if and only if WAP(A∗)∗ is φ-contractible.

Example 4.3. In [8, Example 2.5], it is shown that ℓ1(Z+) is φz-amenable when|z| = 1. Then, Theorem 4.1 yields that WAP(ℓ1(Z+)∗)∗ is φz-contractible. Notice thatWAP(ℓ1(Z+)∗)∗ is not contractible (or even Connes amenable as ℓ1(Z+) is not Connesamenable by [2, Theorem 5.13]).

Example 4.4. Let G be a locally compact group and let A(G) and VN(G)(= A(G)∗)be the Fourier algebra and the von Neumann algebra of G, respectively. Hence, byTheorem 4.1, WAP(VN(G))∗ is φt-contractible for every t ∈ G.

Theorem 4.5. Let A be a Banach algebra, and let φ ∈ ∆(A). Then the following areequivalent:

(i) WAP(A∗)∗ is φ-contractible.

128

Dual Banach algebras; φ-contractibility

(ii) Whenever π : A −→ L(E) is a continuous representation on a reflexive Banachspace E, there exists a φ-quasi expectation Q : L(E) −→ π(A)φ.

Theorem 4.6. Let φ be the augmentation character on ℓ1(N∧). Then WAP(ℓ1(N∧)∗)∗

is not φ-contractible.

Corollary 4.7. WAP(ℓ1(N∨)∗)∗ is not 0-contractible.

Example 4.8. Let φ be the augmentation character on ℓ1(N∧). Then ℓ1(N∧) is notφ-amenable.

Example 4.9. By Theorem 4.1 and corollary 4.7, ℓ1(N∨) is not 0-amenable.

[1] H. G. Dales, Banach algebras and automatic continuity, Clarendon Press, Oxford, 2000.[2] M. Daws, Connes-amenability of bidual and weighted semigroup algebras, Math. Scand. 99

(2006), 217-246.[3] M. Daws, Dual Banach algebras: representations and injectivity, Studia Math. 178 (2007), 231-

275.[4] M. Essmaili, M. Filali, φ-amenability and character amenability of some classes of Banach

algebras, Houston J. Math. 39 (2013), 515-529.[5] A. YA. Helemskii, Some remarks about ideas and results of topological homology In: R. J. Loy

(ed.), Conference on automatic continuity and Banach algebras, pp. 203-238. Australian NationalUniersity, 1989.

[6] Z. Hu., M. S. Monfared, T. Traynor, On character amenable Banach algebras, Studia Math. 193(2009), 53-78.

[7] B.E. Johnson, Cohomology in Banach algebras, Mem. Amer. Math. Soc. 127 (1972).[8] E. Kaniuth, A. T. Lau, J. Pym, On φ-amenability of Banach algebras, Math. Proc. Camb. Phil.

Soc. 144 (2008), 85-96.[9] E. Kaniuth, A. T. Lau, J. Pym, On character amenability of Banach algebras, J. Math. Anal. Appl

344 (2008), 942-955.[10] A. Mahmoodi, Approximate injectivity of dual Banach algebras, Bull. Belgian Math. Soc. Simon

Stevin. 20 (2013), 831-842.[11] A. Mahmoodi, Connes-amenability-like properties, Studia Math. 220 (2014), 55-72.[12] A. Mahmoodi, On φ-Connes amenabilityof dual Banach algebras, J. Linear. Topological Algebra.

3 (2014), 211-217.[13] A. Mahmoodi, Bounded approximate Connes-amenability of dual Banach algebras, Bull. Iranian

Math. Soc. 41 (2015), 227-238.[14] M. S. Monfared, Character amenability of Banach algebras, Math. Proc. Camb. Phil. Soc. 144

(2008), 697-706.[15] V. Runde, Amenability for dual Banach algebras, Studia Math. 148 (2001), 47-66.[16] V. Runde, Lectures on amenability, Lecture Notes in Mathematics 1774, Springer Verlag, Berlin,

2002.[17] V. Runde, Dual Banach algebras: Connes-amenability, normal, virtual diagonals, and injectivity

of the predual bimodule, Math. Scand. 95 (2004), 124-144.

A. Mahmoodi,Department of Mathematics, Central Tehran Branch, Islamic Azad University ,Tehran, Irane-mail: [email protected]

129



January 20–21, 2016

Trivolutions in algebras related to second duals of hypergroupalgebras

A. R. Medghalchi and R. Ramazani∗

Abstract

Let K be a hypergroup. In this paper, we study trivolutions on the Banach algebra Lc(K)∗∗, equipped withits first Arens product and we show that if K is a non-discrete hypergroup, then there are no trivolutionson Lc(K)∗∗. We also find a necessary and sufficient conditions for Lc(K)∗∗ to admit an involution.

2010 Mathematics subject classification: 43A20, 43A22.Keywords and phrases: Hypergroup, second dual algebra, involution.

1. introduction and preliminaries

In [2], Filali, Sangani Monfared and Singh defined a trivolution on a complex algebraA as a non-zero conjugate-linear, anti-homomorphism on A, which is a generalizedinverse of itself, that is, τ3 = τ and showed that The algebra L∞0 (G)∗ ( L∞0 (G) is theclosed ideal of L∞(G) consisting of all f ∈ L∞(G) such that for given ϵ > 0, thereexists a compact set K ⊂ G such that ∥ f ∥G\K < 0) has an involution if and only if G isdiscrete. Further, if G is discrete, L∞(G)∗ has a trivolution with range L1(G), extendingthe natural involution on L1(G) [[2], Theorem 4.4].The aim of this paper is to generalize the above results to very general hypergroupalgebras which include not only group algebras but also most semigroup algebras .For a locally compact Hausdorff space K let M(K) be the Banach space of all boundedcomplex regular measures on K. For all x ∈ K, δx denote the unit point mass at x.Suppose K is a locally compact Hausdorff space. Let Mp(K) be the set of allprobability measures on K , Cb(K) be the Banach space of all continuous boundedcomplex-valued functions on K. We denote by C0(K) the space of all continuousfunctions on K vanishing at infinity and by Cc(K) the space of all continuous functionson K with compact support . The space K is called a hypergroup if there is a mapλ : K × K −→ Mp(K) with the following properties:(i) for every x, y ∈ K, the measures λ(x,y) (the value of λ at (x, y)) have a compact

support ;

∗ speaker

130

A. R. Medghalchi and R. Ramazani

(ii) for each ψ ∈ Cc(K), the map (x, y) 7−→∫

K ψ(t)dλ(x,y)(t) is in Cb(K × K) andx 7−→

∫K ψ(t)dλ(x,y)(t) is in Cc(K) for every y ∈ K;

(iii) the convolution (µ, ν) 7−→ µ ∗ ν of measures defined by∫Kψ(t)d(µ ∗ ν)(t) =

∫K

∫K

∫Kψ(t)dλ(x,y)(t)dµ(x)dν(y),

is associative where µ, ν ∈ M(K), ψ ∈ C0(K) ( note that λ(x,y) = δx ∗ δy);(iv) there is a unique point e ∈ K such that λ(x,e) = δx for all x ∈ K.When λ(x,y) = λ(y,x) , we say that K is a commutative hypergroup. With this definitionM(K) becomes a Banach algebra. A positive measure m on K is called a left invariantmeasure (Haar measure) if for all x ∈ K, δx ∗ m = m. We do not assume that K hasa Haar measure, we define L(K) = µ | µ ∈ M(K), x 7→ |µ| ∗ δx, x 7→ δx ∗ |µ| arenorm - continuous, which is also an ideal in M(K) and also let K is foundation, i.e.K = cl(

∪µ∈L(K) suppµ), L(K) admits a bounded approximate identity [[4], Lemma 1].

We recall the definition of the first (left) Arens product on the second dual A∗∗ of aBanach algebra A. For f ∈ A∗, a ∈ A, let f .a ∈ A∗ be defined by

⟨ f .a, b⟩ = ⟨ f , ab⟩, b ∈ A.

Now for G ∈ A∗∗ and f ∈ A∗, let G. f ∈ A∗ be defined by

⟨G. f , a⟩ = ⟨G, f .a⟩, a ∈ A.

Finally, for F,G ∈ A∗∗ let FG ∈ A∗∗ be defined by

⟨FG, f ⟩ = ⟨F,G. f ⟩, f ∈ A∗.

We will also need the following definition of the second (right) Arens product. Withthe same notation as above we define a. f ∈ A∗, f .F and FG respectively by

⟨a. f , b⟩ = ⟨ f , ba⟩, b ∈ A,

⟨ f .F, a⟩ = ⟨F, a. f ⟩, a ∈ A,

⟨FG, f ⟩ = ⟨G, f .F⟩, f ∈ A∗.

Let B = L(K)∗L(K). In [5] it has been shown that B∗ (dual of B) is a Banach algebra byan Arens type product and that L(K) ⊆ B∗. Also if K admits an invariant measure(Haarmeasure) m, then B = LUC(K), where LUC(K) = f | f ∈ Cb(K), x → lx f from Kinto Cb(K) is continuous [[6], Proposition 2.4]. Most of our notations in this papercome from [5], where E ∈ L(K)∗∗ is the weak*-limit of (eα), a bounded approximateidentity in L(K) and E is in fact a right identity for L(K)∗∗. Also ε(K) = π−1(δe) andε1(K) = E ∈ ε(K) | ∥E∥ = 1, where π : L(K)∗∗ −→ B∗ is the adjoint of embedding ofB in L(K)∗.Following [[5], Definition 8] a compact set Y ⊆ K is called a compact carrier form ∈ L(K)∗∗, if for all f ∈ L(K)∗

⟨m, f ⟩ = ⟨m, fχY⟩,where fχY is defined by ⟨ fχY , µ⟩ = ⟨ f , χYµ⟩, for all µ ∈ L(K). Now let

Lc(K)∗∗ = clL(K)∗∗m|m ∈ L(K)∗∗,m has compact carrier.

131

Trivolutions in algebras related to second duals of hypergroup algebras

2. main results

Throughout this paper K is a foundation hypergroup. A trivolution on a complexalgebra A is a non-zero, conjugate linear, anti-homomorphism τ : A −→ A, such thatτ3 = τ τ τ = τ. When A is a normed algebra, we shall assume that ∥τ∥ = 1. Thepair (A, τ) will be called a trivolutive algebra ( see [2]). It follows from the definitionthat every involution on a non-zero complex algebra is a trivolution. Conversely, atrivolution which is either injective or surjective, is an involution ( see [2]).

Let K be a non-discrete hypergroup and X and Y be subalgebras of L(K)∗∗ suchthat Lc(K)∗∗ ⊆ Y ⊆ X. Then there are no trivolutions of X onto Y . In particular,L(K)∗∗ has no trivolutions with range Lc(K)∗∗. A involution on a hypergroup K is ahomeomorphism x 7→ x in K such that ˜x = x and e ∈ suppλ(x,x) for all x ∈ K. For eachµ ∈ M(K) define µ ∈ M(K) byµ(A) = µ(A) i.e.∫

Kf (x)dµ(x) =

∫K

f (x)dµ(x), ( f ∈ Cc(K))

then µ −→ µ is an involution on M(K) such that M(K) and L(K) are Banach *- algebras[3] and λ(x,y) = λ(y,x) for all x, y ∈ K [1]. Let K be a compact hypergroup with aninvolution. Then for each E ∈ ε(K), there are trivolutions of L(K)∗∗ onto EL(K)∗∗.Let K be a hypergroup with an involution. Then for each E ∈ ε1(K), there exists atrivolution of Lc(K)∗∗ onto ELc(K)∗∗. Now we state our other main result. Let Kbe a hypergroup whit an involution. Then Lc(K)∗∗ has an involution if and only if G isdiscrete.

Let K be a hypergroup whit an involution If Lc(K)∗∗ is amenable, then Lc(K)∗∗ hasan involution.

Let G be a locally compact group. Then Lc(G)∗∗ = L∞0 (G)∗, where L∞0 (G) is theclosed ideal of L∞(G) consisting of all f ∈ L∞(G) such that for given ϵ > 0, thereexists a compact set K ⊂ G such that ∥ f ∥G\K < ϵ. By theorem 2 L∞0 (G)∗ admits aninvolution if and only if G discrete.

[1] C. F. Dunkl, The measure algebra of a locally compact hypergroup, Trans. Amer. Math. Sot. 179(1973), 331-348.

[2] M. Filali, M. Sangani Monfared and A. I. Singh, Involutions and trivolutions in algebras relatedto second duals of group algebras, Illinois J. Math., Volume 57, Number 3 (2013), 755-773.

[3] R. I. Jewett, Spaces with an abstract convolution of measures. Adv. Math.18 , 1U101 (1975).[4] F. Ghahramani and A. R. Medghalchi, Compact multipliers on hypergroup algebras, Math. Proc.

Camb. Phil. Soc. 98, 493U500 (1985).[5] A. R. Medghalchi, The second dual of a hypergroup, Math. Z.210 (1992), 615U624.[6] A.R. Medghalchi, Cohomology on hypergroup algebras. Studia Scientiarum Mathematicarum,

Hungarica.39 (2002), 297-307.[7] A. R. Medghalchi and S. M. S. Modarres, Amenability of the second dual of hypergroup algebras,

Acta Math. Hungar. 86 (4) (2000), 335-342.

132

A. R. Medghalchi and R. Ramazani

A. R. Medghalchi,Department of Mathematics,Kharazmi University,Taleghani Avenue, 15618,Tehran, Iran.e-mail: [email protected]

R. Ramazani,Department of Mathematics, Kharazmi University,Taleghani Avenue, 15618,Tehran, Iran.e-mail: [email protected]

133



January 20–21, 2016

Ternary n-Weak Amenability of C∗-Algebras and Group Algebras

M. R. Miri and M. Niazi∗

AbstractWe introduce the notion of ternary n-weak amenability for every n ∈ N, in the context of triple systems andprove that every group algebra of a discrete abelian group and every commutative unital C∗-algebras areternary n-weakly amenable. These results present a somehow unified extension of the previous ternaryweak amenability results and n-weak amenability results in the category of triple systems and Banachalgebras, respectively.

2010 Mathematics subject classification: Primary 17C65, Secondary 46H25..Keywords and phrases: Jordan triple, ternary module, ternary derivation, weak amenability.

1. Introduction

In 2013 A. M. Peralta et al. [2] introduced a ternary module structure in the categoryof Jordan triple systems. With this proposed structure it turns out that the dual of aBanach Jordan triple can be endowed a ternary module structure and this prepared agroundwork to study the weak amenability of triple systems. They showed that everycommutative C∗-algebra is ternary weakly amenable and in a more general frameworkthey showed that every commutative JBW∗-triple is ternary weakly amenable. Thiswas a counterpart result in the category of triple systems to the result of Haagerupin the category of binary associative algebras which shows that every C∗-algebra isweakly amenable.

In 1988 H. G. Dales et al. [1] strengthen the result of Haagerup by introducing thenotion of n-weak amenability (n ∈ N), for a Banach algebra and proving that everyC∗-algebra is n-weakly amenable for any integer n. It is reminded that for an integern a Banach algebra A is called n-weakly amenable if every bounded derivation from Ainto its iterated dual, A(n), is inner.

In this paper we establish counterpart results to the Dales’s results, in the categoryof triple systems. To this end we need to improve the notion of ternary modules insuch a way that make it possible to endow every iterated dual spaces of a BanachJordan triple a ternary module structure. It should be noticed that with the proposeddefinition of a ternary module by A. M. Peralta et al. it is impossible to endow the∗ speaker

134

M. R. Miri andM. Niazi

bidual of Banach Jordan triple a module structure in a natural way that contains E as asubmodule.

In the next section we recall the definition and some basic properties of Jordantriples and define a new type of ternary modules, ternary derivations and introducethe notion of ternary n-weak amenability. In section 3 we study n-weak amenabilityresult for Banach ∗-algebras and conclude that every commutative unital C∗-algebrasand every group algebras, considered as Banach Jordan triples, are ternary n-weaklyamenable.

2. Ternary Modules

To investigate ternary n-weak amenability problem in the context of Jordan tripleswe present a new type of ternary modules over Jordan triples. Definition and detailson JB∗-triples can be found in [4].

2.1. Ternary Modules In [2] A. M. Peralta et al. introduced a ternary modulestructure which intended to explore the weak amenability problem for Jordan triples.To explore the n-weak amenability problem in the context of Jordan triples we need tointroduce another type of ternary module. We call the previously defined one a ternarymodule of type (I) and call the new one a ternary module of type (II):

Definition 2.1. Let E be a Jordan triple and X be a complex vector space. Considerthe following mappings and axioms:

π1 : X × E × E → X, π1(x, a, b) = [x, a, b]1,

π2 : E × X × E → X, π2(a, x, b) = [a, x, b]2,

π3 : E × E × X → X, π3(a, b, x) = [a, b, x]3.

(1) π1 is linear in the first and second variables and conjugate linear in the thirdvariable. π2 is conjugate linear in each variable. π3 is conjugate linear in thefirst variable and linear in the second and third variables.

(1)′ Each of the mappings π1, π2 and π3 is linear in the first and third variables andconjugate linear in the second variable.

(2) [x, b, a]1 = [a, b, x]3, and [a, x, b]2 = [b, x, a]2 for every a, b ∈ E and x ∈ X.

(3) Let [·, ·, ·] denote any of the mappings [·, ·, ·]1, [·, ·, ·]2, [·, ·, ·]3 or the tripleproduct of E. Then the identity

[a, b, [c, d, e]] = [[a, b, c], d, e] − [c, [b, a, d], e] + [c, d, [a, b, e]],

holds for every a, b, c, d, e where one of them is in X and the other ones are in E.

When the mappings π1, π2 and π3 satisfy the axioms (1), (2) and (3), X is called aternary E-module of type (I) and when they satisfy the axioms (1)′, (2) and (3), X iscalled a ternary E-module of type (II).

135

Ternary n-Weak Amenability

We usually write the expression “ternary E-module”, without declaring the type,when a statement is true for both types or when the type is clear from the context.

When E is a Banach Jordan triple, X is a Banach space and the module actionsπ1, π2 and π3 are continuous we say that X is a Banach ternary E-module.

To simplify notations, hereafter, the module actions [·, ·, ·]1, [·, ·, ·]2, [·, ·, ·]3 andthe triple product of E will be denoted by [·, ·, ·] and its meaning will be clear from thecontext.

Let E be a Jordan triple and X be a ternary E-module. We define the followingmappings

[θ, a, b]1(x) = θ[a, b, x]3, [a, θ, b]2(x) = θ[a, x, b]2, [a, b, θ]3(x) = θ[x, a, b]1 (1)

where a, b ∈ E, x ∈ X and θ ∈ X∗. The next theorem shows that the above mappingsoperate as module actions for the dual space X∗.

Theorem 2.2. Let E be a Jordan triple. If X is a ternary E-module of type (I) (resp.(II)) then its dual space X∗ is a ternary E-module of type (II) (resp. (I)).

It is easy to see that every Banach Jordan triple E is a Banach ternary E-module oftype (II) under its own ternary multiplication (as module actions).

Let E be a Jordan triple, then Theorem 2.2 confirms that E∗ is a Banach ternaryE-module of type (I), E∗∗ is a Banach ternary E-module of type (II). This means thatthe iterated dual E(n) is a Banach ternary E-module of type (I) whenever the integer nis odd and is a Banach ternary E-module of type (II) whenever n is even.

2.2. Ternary Derivations Regarding the two different types of ternary modules it isnatural to have two different types of derivations:

Definition 2.3. A ternary derivation from a Jordan triple E into a ternary E-moduleof type (I) (resp. (II)) X is a conjugate linear (resp. linear) mapping D : E → X,satisfying

D[a, b, c] = [Da, b, c] + [a,Db, c] + [a, b,Dc],

for every a, b, c in E.

Let E be a Banach Jordan triple and X be a Banach ternary E-module. Applyingthe axiom (3) of Definitions 2.1, for every b ∈ E and x ∈ X, we see that the mappingδ(b, x) : E → X, defined by

δ(b, x)(a) = [b, x, a] − [x, b, a], (a ∈ E) (2)

is a ternary derivation. A finite sum of the above derivations (2) is called a ternaryinner derivation.

A Banach Jordan triple E is said to be ternary amenable if every ternary derivationfrom E into the dual of a Banach ternary E-module is a ternary inner derivation.For any integer n ∈ N, we say that E is ternary n-weakly amenable if every ternaryderivation from E into the iterated dual, E(n) is a ternary inner derivation. We simply

136

M. R. Miri andM. Niazi

write ternary weakly amenable instead of ternary 1-weakly amenable. In 2013, A. M.Peralta et al. [2] showed that every commutative C∗-algebra, and particularlly, everycommutative JBW∗-triple is ternary weakly amenable.

3. Ternary n-Weak Amenability of ∗-Algebras

Let A be a Banach ∗-algebra. The following triple product:

[a, b, c] =12

(ab∗c + cb∗a), (a, b, c ∈ A)

provides A with a ternary structure. Then for every n ∈ N, the iterated dual spaces A(n)

are turned out to be ternary A-modules by the rule described after Theorem 2.2. Weremind that the iterated dual spaces A(n) also enjoy binary A-module structures by thefollowing recursively defined module actions:

(aθ)(φ) = θ(φa), (θa)(φ) = θ(aφ)

where a ∈ A, θ ∈ A(n) and φ ∈ A(n−1). We also define recursively an involution ∗ onA(n), by

θ∗(φ) = θ(φ∗), (θ ∈ A(n), φ ∈ A(n−1)).

The following proposition determines the relationship between ternary moduleactions and binary module actions on the iterated duals of a Banach ∗-algebra.

Proposition 3.1. Let A be a Banach ∗-algebra and n ∈ N. For every a, b ∈ A andθ ∈ A(2n−1), we have

[θ, a, b] = [b, a, θ] =12

(θab∗ + b∗aθ), [a, θ, b] =12

(a∗θ∗b∗ + b∗θ∗a∗)

and for every θ ∈ A(2n)

[θ, a, b] = [b, a, θ] =12

(θa∗b + ba∗θ), [a, θ, b] =12

(aθ∗b + bθ∗a).

The following theorem inspired by the results in [2, Section 3]; see also [3].

Theorem 3.2. Let n ∈ N. A commutative unital Banach ∗-algebra is ternary n-weaklyamenable whenever it is n-weakly amenable.

It has been known that all C∗-algebras and also the convolution group algebraL1(G), a locally compact group G, are n-weakly amenable for each n ∈ N; see [1, 5].These observations together with the above theorem lead us to the next result providingternary n−weak amenability for certain C∗-algebras and L1(G).

Corollary 3.3. (i) Every commutative unital C∗-algebra is ternary n-weakly amenablefor every n ∈ N.

(ii) For every discrete abelian group G, the convolution group algebra L1(G) isternary n-weakly amenable for each n ∈ N.

137

Ternary n-Weak Amenability

Acknowledgement

The authors would like to thank H.R. Ebrahimi Vishki for his suggestions andcomments.

[1] H. G. Dales, F. Ghahramani and N. Grønbæk, Derivations into iterated duals of banach algebras,Studia Math. 128(1) (1998) 19-54.305-319.

[2] T. Ho, A. M. Peralta and B. Russo, Ternary weakly amenable C∗-algebras and JB∗-triples, Quart.J. Math. 64 (2013) 1109-1139.

[3] M.R. Miri, M. Niazi and H.R. Ebrahimi Vishki Lifting ternary derivations and ternary n−weakamenability of bidual space, Preprint (2015).

[4] B. Russo, Structure of JB∗-triples, In: Jordan Algebras, Proceedings of the Oberwolfach Confer-ence 1992, Eds: W. Kaup, K. McCrimmon, H. Petersson, de Gruyter, Berlin (1994) 209-280.

[5] Y. Zhang, 2m-Weak amenability of group algebras, J. Math. Anal. Appl. 396 (2012) 412-416.

M. R. Miri,Department of Mathematics,University of Birjand,Birjand, Irane-mail: [email protected]

M. Niazi,Department of Mathematics,University of Birjand,Birjand, Irane-mail: [email protected]

138



January 20–21, 2016

On local information functional for discrete dynamical systems

U. Mohammadi

Abstract

In this paper, the concept of information functional for discrete dynamical systems on compact metricspaces is presented using the generator notion. Also, the independence of information functional ofgenerators is proved. Moreover, the invariance of the information functional of a dynamical system,under topological isomorphism, is deduced.

2010 Mathematics subject classification: Primary 37A35, Secondary 28D20.Keywords and phrases: dynamical system, generator, information functional.

1. Introduction

Shannon [3] introduced the concept of information function and investigated someproperties of this function. McMillan , Dumitrescu and Tok [4] studied some prop-erties of the fuzzy information function. Recently, Guney, Tok and Yamankaradenizdefined fuzzy local information function and stated some properties of this functionin [1]. In this paper, a new notion of information functional for discrete dynamicalsystems on compact metric spaces is presented using the generator notion and its prop-erties are investigated. Also, it is shown that the information functional of dynamicalsystems is invariant under isomorphism.

2. Preliminary facts

Let (X, β) denotes a σ−finite measure space, i.e. a set equiped with a σ−algebraβ of subsets of X. Further let µ denote a probability measure on (X, β). Then(X, β, µ) is called a probability space. Let T : X → X be a measure preservinginvertible transformation of probability space (X, β, µ). In particular T (β) = β andµ(T−1(A)) = µ(A) for all A ∈ β. Then (X, β, µ, T ) is called a dynamical system.In this article the set of all probability measures on X preserving T is denoted byM(X,T ). We also write E(X,T ) for the set of all ergodic measures of T .

Definition 2.1. A partition ξ is a refinement of a partition η, if every element of η is aunion of elements of ξ and it is denoted by η ≺ ξ.

139

U. Mohammadi

Definition 2.2. Given two partitions ξ, η their common refinement is defined as fol-lows:

ξ ∨ η = Ai ∩ B j; Ai ∈ ξ, B j ∈ η.In the following we recall some classical results that we need in the sequel.

Theorem 2.3. (Choquet) Suppose that Y is a compact convex metrisable subset ofa locally covex space E, and x0 ∈ Y. Then, there exists a probability measureτ on Y which represents x0 and is supported by the extreme points of Y, that is,Φ(x0) =

∫Y Φdτ for every continuous linear functional Φ on E, and τ(ext(Y)) = 1.

Proof. See [2].

Let µ ∈ M(X,T ) and f : X → R be a bounded measurable function. As we knowthat E(X,T ) equals the extreme points of M(X,T ), applying the Choquets Theoremfor E = M(X), the space of finite regular Borel measures on X, and Y = M(X,T ),and using the linear functional Φ : M(X) → R given by Φ(µ) =

∫X f dµ, we have the

following Corollary:

Corollary 2.4. Suppose that T : X → X is a continuous map on the compact metricspace X. Then, for each µ ∈ M(X,T ), there is a unique measure τ on the Borel subsetsof the compact metrsable space M(X,T ), such that τ(E(X,T )) = 1 and∫

Xf (x)dµ(x) =

∫E(X,T )

(∫

Xf (x)dm(x))dτ(m)

for every bounded measurable function f : X → R.

Under the assumptions of Corollary 2.4, we write µ =∫

E(X,T ) mdτ(m), called theergodic decomposition of µ.

3. Information functional for continuous dynamical systems

Definition 3.1. Suppose that T : X → X is a continuous map on the topological spaceX, x ∈ X and A a Borel subset of X. Then

mx(A) = lim supn→∞

1n

cardk ∈ 0, 1, ..., n − 1 : T k(x) ∈ A.

Now, let x ∈ X and ξ = A1, A2, ..., An and η = B1, B2, ..., Bm be finite Borel partitionsof X. We define

I∗(x,T, ξ) := −n∑

i=1

χAi (x) log mx(Ai);

and

I∗(x,T, ξ|η) := −∑i, j

χ(Ai∩B j)(x) logmx(Ai ∩ B j)

mx(B j).

(We assume that log 0 = −∞ and 0 ×∞ = 0).

140

Information functional

Note that the quantity I∗(x,T, ξ|η) is the conditional version of I∗(x,T, ξ). It is clearI∗(x,T, ξ) ≥ 0.

Theorem 3.2. Suppose that T : X → X is a continuous map on the topological spaceX, x ∈ X and ξ, η, ζ are finite Borel partitions and x ∈ X then(i) I∗(x,T, ξ ∨ η|ζ) = I∗(x,T, ξ|ζ) + I∗(x,T, η|ξ ∨ ζ);(ii) I∗(x,T, ξ ∨ η) = I∗(x,T, ξ) + I∗(x,T, η|ξ);(iii) If ξ ≺ η then I∗(x,T, ξ|ζ) ≤ I∗(x,T, η|ζ);(iv) If ξ ≺ η then I∗(x,T, ξ) ≤ I∗(x,T, η).

Definition 3.3. Suppose that T : X → X is a continuous map on the topological spaceX, x ∈ X and ξ be a finite Borel partition of X. The map I(.,T, ξ) : X → [0,∞] isdefined as

I(x,T, ξ) = lim supl→∞

1l

I∗(x,T,∨l−1i=0T−iξ).

Theorem 3.4. Let ξ be a finite partition of X. Then for every k ∈ N,(i) If ξ ≺ η then I(x,T, ξ) ≤ I(x,T, η);(ii) I(x,T, ξ) = I(x,T,∨k

j=0T− jξ).

Definition 3.5. Let T : X → X is a continuous map on the topological space X. Thena partition ξ of X is called a generator of T if there exists an integer k > 0 such that

η ≺ ∨ki=0T−iξ

for every partition η of X.

Theorem 3.6. Let ξ be a generator of T then I(x,T, η) ≤ I(x,T, ξ), for every partitionη of X.

Definition 3.7. Suppose that T : X → X is a continuous map on the compact metricspace X, and ξ be a generator for the dynamical system (X,T ). Let µ ∈ M(X,T ). Theinformation functional of T ( with respect to µ), is defined as

Iµ(T, ξ) =∫

XI(x,T, ξ)dµ(x)

In the following, we will prove the independence of information functional fromthe selection of the generator.

Theorem 3.8. Definition 3.7 is independent of the choice of generator i.e if ξ and η aretwo generators of T then,

Iµ(T, ξ) = Iµ(T, η).

Remark 3.9. By Theorem 3.8, we conclude that the definition of information functionalis independent of the selection of generators. Therefore, given any invariant measureµ and any relative generator ξ, we have the unique information functional. So, we canwrite Iµ(T ) for Iµ(T, ξ) without confusion.

141

U. Mohammadi

Example 3.10. Let X = [0, 1] and T : X → X be the doubling map T (x) = 2x(mod1).If we equip X = [0, 1] with the Borel sigma-algebra, then T is easily seen to preserve

Lebesgue measure µ. Let ξ = [0, 12

), [12, 1); then observe that

ξ ∨ T−1ξ = [0, 14

), [14,

12

), [12,

34

), [34, 1).

and more generally,

∨l−1i=0T−iξ = [ i

2n ,i + 12n ) : i = 0, 1, ..., 2l − 1.

Thus ξ is a generator and for each x ∈ X we can now calculate

I∗(x,T,∨l−1i=0T−iξ) = −

2l−1∑i=0

µ([i2l ,

i + 12l )) log µ([

i2l ,

i + 12l ))

= −2l−1∑i=0

(12l ) log(

12l )

= −2l(12l ) log(

12l )

= l log 2.

Thus we see that1l

I∗(x,T,∨l−1i=0T−iξ) = log 2 and thus letting l → ∞ gives that

I(x,T, ξ) = log 2. So, we have Iµ(T, ξ) = log 2.

Theorem 3.11. Suppose that T : X → X is a continuous map on the compact metricspace X. Then the map µ→ Iµ(T ) is affine.

Definition 3.12. We say that two dynamical systems (X,T1) and (Y,T2) are isomorphicif there exists a homeomorphism φ : X → Y such that φoT1 = T2oφ.

Theorem 3.13. Suppose that T : X → X is a continuous map on the compactmetric space X. If two dynamical systems (X,T1) and (Y,T2) are isomorphic, andµ ∈ M(X,T ), then,

Iµ(T1) = Iµ(T2).

Theorem 3.14. Suppose that T : X → X is a continuous map on the compact metricspace X. If µ ∈ M(X,T ) and µ =

∫E(X,T ) mdτ(m) is the ergodic decomposition of µ,

then,

Iµ(T ) =∫

E(X,T )Im(T )dτ(m).

[1] I. Guney, I. Tok andM. Yamankaradeniz, On the fuzzy local information function, Balkan Journalof Mathematics, 1 (2013) 44-60.

142

Information functional

[2] R. Phelps, Lectures on Choquets Theorem, D. Van Nostrand Co., Inc., Princeton, N. J.-Toronto,Ont.-London, 1966.

[3] C. Shannon, A mathematical theory of communication. Bell Syst Tech Journal, 27 (1948), 379-423.

[4] I. Tok, On the fuzzy information function, Doga Tr. J. Math., 10 (1986), 312-318.[5] P. Walters, An Introduction to Ergodic Theory, Springer Verlag, Berlin, 1982.

U. Mohammadi,Department of Mathematics, Faculty of Science,University of Jiroft,Jiroft, Irane-mail: [email protected]

143



January 20–21, 2016

On some solvable extensions of the Heisenberg group

M. Nasehi

Abstract

In this paper we consider two families of four or five- dimensional Riemannian solvable Lie groupswhich are one and two-dimensional extensions of the Heisenberg group. We first completely determineleft-invariant harmonic vector fields on these spaces. Then we investigate left-invariant Ricci solitonsand Einstein-like metrics on these homogeneous spaces. Later on, in the case that these spaces are four-dimensional we build all five-dimensional homogeneous Sasakian spaces obtained by the contactizationof these spaces.

2010 Mathematics subject classification: Primary 53C30 Secondary 53C15.Keywords and phrases: Solvable Lie groups, Heisenberg group, Harmonic vector fields, Contactization.

1. Introduction

Heisenberg groups were introduced by Kaplan in 1981. These spaces play an impor-tant role in geometric analysis, mathematical physics and Lie groups. For a surveyabout the geometry of generalized Heisenberg groups we refer to [4]. Two families offour or five dimensional solvable Lie groups which are extensions of the Heisenberggroup are A4(λ, µ) and A5(λ, µ, ν), where λ and µ are positive numbers. Homogeneousstructures and reductive decompositions of these spaces are investigated in [3]. Ouraim in this paper is to investigate some other geometrical properties of these spaces.In fact in Section 2 we first recall the curvature tensor components in [3] and obtainthe Ricci tensor components on these spaces. In Section 3 we completely obtain left-invariant harmonic vector fields on these spaces. In Section 4 we first investigate left-invariant Ricci solitons on these spaces. Then we prove the existence of one or twodimensional extensions of the Heisenberg group with parallel Ricci tensor. In Section5 we obtain all 5-dimensional homogeneous contact spaces which are obtained by thecontactization of Kähler spaces (A4(λ, µ), gλ,µ, J).

2. Curvature of some solvable extensions of the Heisenberg group

In this section we consider two families of one or two dimensional extensions ofthe Heisenberg group which are given in [3] and obtain Ricci tensor components onthese spaces as follows

144

M. Nasehi

The one-dimensional extensions of the Heisenberg group: These spaces are four-dimensional and denoted by A4(λ, µ), where λ, µ > 0 and the non-zero brackets aregiven by [X1, X2] = λX3, [X4, X1] = µX1, [X4, X2] = µX2, [X4, X3] = 2µX3, whereX1, . . . , X4 is an orthonormal basis of the Lie algebra A4(λ, µ). Also the non-zeroRiemannian curvature components are given

R1212 = −(3λ2

4+ µ2),R1234 = R3412 = λµ,R1313 = R2323 = (

λ2

4− 2µ2),

R3434 = −4µ2,R1414 = R2424 = −µ2, R1432 = R2341 =λµ

2, (1)

and the ones obtained by them using the symmetries of the curvature tensor. Further-more the left-invariant metric gλ,µ on these spaces is given by

gλ,µ = e−4µx4(e2µx4 +λ2

4x2

2)dx12 + (e2µx4 +

λ2

4x1

2)dx22 + dx3

2

− λ2

2x1x2dx1dx2 + λ(x2dx1dx3 − x1dx2dx3) + dx4

2.

see [3]. Also, by using the Ricci tensor formula ρi j := ρeie j =∑

t

< R(ei, Xt)e j, Xt >,

we can obtain the non-zero Ricci tensor components as follows

ρ11 = ρ22 = −(λ2

2+ 4µ2), ρ33 =

λ2

2− 8µ2, ρ44 = −6µ2, (2)

These components imply that the scalar curvature τ is equal to τ = −22µ2 − λ2

2 .The two-dimensional extensions of the Heisenberg group: These spaces are five-dimensional and denoted by A5(λ, µ, ν), where λ, µ > 0 and the non zero brackets aregiven by [X1, X2] = λX3, [X4, X1] = µX1, [X4, X2] = µX2, [X4, X3] = 2µX3, [X4, X5] =νX5, where X1, . . . , X5 is an orthonormal basis for the Lie algebra A5(λ, µ, ν). Alsothe non-zero curvature tensor components are given by equations (1) and R1551 =

νµ,R2525 = −νµ,R3535 = −2µν,R4545 = −ν2 and the ones implied by them using thesymmetries of the curvature tensor. Furthermore the left-invariant metric on A5(λ, µ, ν)is given by

gλ,µ,ν = e−4µx4(e2µx4 +λ2

4x2

2)dx12 + (e2µx4 +

λ2

4x1

2)dx22 + dx3

2+

dx42 − λ

2

2x1x2dx1dx2 + λ(x2dx1dx3 − x1dx2dx3) + e−2νx4 dx5

2.

see [3]. Also, the non zero Ricci tensor components can be described by ρ11 = ρ22 =

−( λ2

2 + 4µ2 + µν), ρ33 =λ2

2 − 8µ2 − 2µν, ρ44 = −6µ2 − ν2, and ρ55 = −4µν − ν2. whichimply that the scalar curvature τ is equal to τ = −18µ2 − 6µν − ν2.

145


3. Left-invariant harmonic vector fields on some solvable extensions of theHeisenberg groups

To investigate left-invariant harmonic vector fields on two families of four or fivedimensional Riemannian solvable Lie groups A4(λ, µ) and A5(λ, µ, ν), we recall somefacts and for more details we refer to [1, 6]. Let (M, g) be an oriented n-dimensionalRiemannian smooth manifold and (T M, gs) be its tangent bundle with the Sasakianmetric gs. Then the energy of the smooth vector field X : (M, g) → (T M, gs) can bedefined by E(X) = n

2 vol(M, g)+ 12

∫M ∥ ∇X ∥2 dv.Also the critical points for the energy

function E|χP(M), restricted to χP(M), where χP(M) = W ∈ χ(M) : ∥W∥2 = ρ2 andρ , 0 is a real constant, can be obtained by X such that ∇∗∇X is collinear to X, where

∇∗∇X =n∑

i=1

(∇Xi∇Xi X − ∇∇Xi Xi X), (3)

and X1, . . . , Xn is an orthonormal basis on (M, g). By the Euler-Lagrange equationsthese vector fields are called harmonic vector fields.

Theorem 3.1. Let (G, g) be one of the two families of four or five dimensional Rie-mannian solvable Lie groups A4(λ, µ) and A5(λ, µ, ν). Then a left-invariant harmonicvector field X on these spaces can be described by one of the following forms(a) If (G, g) is (A4(λ, µ), gλ,µ), then X = K1X1+K2X2, X = K3X3, X = K4X4 and X = 0,where K1, . . . ,K4 are some real constants.(b) If (G, g) is (A5(λ, µ, ν), gλ,µ,ν), then X = K1X1 + K2X2, X = K3X3, X = K4X4,X = K5X5 and X = 0, where K1, . . . ,K5 are some real constants.

Proof. We will prove the result for A4(λ, µ) and analogy we have the result forA5(λ, µ, λ). Assume that X = K1X1+K2X2+K3X3+K4X4 be an arbitrary left-invariantvector field on A4(λ, µ). Then by (3) we obtain that ∇∗∇X = −( λ

2

2 +µ2)(K1X1+K2X2)−

( λ2

2 + 2µ2)K3X3 − 6µ2K4X4. Suppose that δ ∈ R, then from the condition ∇∗∇X = δXwe obtain the result.

4. Left-invariant Ricci solitons and Einstein-like metrics on some solvableextensions of the Heisenberg group

Let G be a Lie group and g be a left-invariant Riemannian metric on G. Then Gis said to be a left-invariant Ricci soliton, if it is equipped with a left-invariant vectorfield T which satisfies the following condition

ρ(s, t) +12

LT g(s, t) = δg(s, t), ∀s, t ∈ G (4)

where ρ is the Ricci tensor, LT is the Lie derivative in the direction of T and δ is a realnumber (see [7]).

Theorem 4.1. Let (G, g) be one of the two families of four or five dimensionalRiemannian solvable Lie groups (A4(λ, µ), gλ,µ) and (A5(λ, µ, ν), gλ,µ,ν). Then G doesnot admit any non zero left-invariant Ricci soliton.

146

M. Nasehi

Proof. Assume that T = K1X1+K2X2+K3X3+K4X4 is an arbitrary left-invariant vectorfield on A4(λ, µ). If we replace (s, t) in Eq.(4) by (X1, X3), (X1, X4) and (X3, X4), thenwe respectively obtain that K2 = K1 = 0 and K3 = 0 which imply that T = K4X4. Alsoif we replace (s, t) in Eq.(4) by (X1, X1), (X3, X3) and (X4, X4), then we respectivelyobtain that δ = −λ

2

2 − 4µ2 −µK4, δ = λ2

2 − 8µ2 − 2µK4 and δ = −6µ2. These imply thatK4 = 0. For A5(λ, µ, ν) we have a similar proof.

Here we recall that a Riemannian manifold (M, g) of dimension n ⩾ 4 is confor-mally flat if and only if we have

RX,Y,Z,W =1

n − 2(g(X,Z)ϱ(Y,W) + g(Y,W)ϱ(X,Z) − g(X,W)ϱ(Y,Z) − g(Y,Z)ϱ(X,W))

− τ

(n − 1)(n − 2)(g(X,Z)g(Y,W) − g(Y,Z)g(X,W)), (5)

Then from the equations (1) and (5) we obtain the following result.

Theorem 4.2. Let (G, g) be one of the two families of four or five dimensionalRiemannian solvable Lie groups (A4(λ, µ), gλ,µ) and (A5(λ, µ, ν), gλ,µ,ν). Then (G, g)is not conformally flat.

Recall that Einstein-like metrics on a Riemannian manifold (M, g) are definedthrough the conditions on the Ricci tensor. In fact a Riemannian manifold (M, g)respectively belong to the classes A, B and P = A ∩ B if and only if its Riccitensor is cyclic-parallel, i.e., ∇XiρX jXk + ∇X jρXkXi + ∇XkρXiX j = 0, is Codazzi tensori.e., ∇XiρX jXk = ∇X jρXiXk and is parallel i.e., ∇XiρX jXk = 0, where Xi, X j and Xk aretangent vectors on M.

Theorem 4.3. (a) All and the ones proper simply connected four dimensional Rieman-nian solvable Lie groups (A4(λ, µ), gλ,µ), equipped with Einstein-like metrics are theones with λ2 = 4µ2, whose Ricci tensor is parallel.(b) All and the ones proper simply connected five dimensional Riemannian solvableLie groups (A5(λ, µ, ν), gλ,µ,ν), equipped with Einstein-like metrics are the ones withλ2 = 4µ2 and ν = 0, whose Ricci tensor is parallel.

Proof. For A4(λ, µ) by (2) the non-zero components of ∇iρ jk are given by ∇1ρ14 =

∇2ρ24 = ∇1ρ41 = ∇2ρ42 = ∇1ρ41 = 2µ3 − λ2µ2 ,∇2ρ13 = −∇1ρ23 = −∇1ρ32 = ∇2ρ31 =

λ3

2 − 2λµ2, ∇3ρ34 = ∇3ρ43 = λ2µ − 4µ3. Then it can be seen that the Ricci tensoron A4(λ, µ) is parallel if and only if λ2 = 4µ2. For A5(λ, µ, ν) by a similar waythe non-zero components of ∇iρ jk are given by ∇1ρ14 = ∇2ρ24 = ∇1ρ42 = ∇2ρ42 =

2µ3 − λ2µ2 − νµ2 + µν2,∇2ρ31 = ∇2ρ13 = −∇1ρ23 = −∇1ρ32 =

λ3

2 −λµν

2 − 2λµ2,∇3ρ43 =

∇3ρ34 = λ2µ − 4µ3 − 4µ2ν + 2µν2,∇5ρ55 = ν

2(8µ + 2ν),∇5ρ45 = −ν2(4µ + ν). Then bya similar way we can see that the Ricci tensor on A5(λ, µ, ν) is parallel if and only ifλ2 = 4µ2 and ν = 0.

147


5. Contactization of (A4(λ, µ), J, gλ,µ)

In [3] it is proved that (A4(λ, µ), J, gλ,µ) with λ = 2µ is a Kähler manifold. Here toobtain the contactization of these spaces, we recall the following facts and for moredetails we refer to [2, 5]. Let ( G

H ,Ω) be a homogeneous symplectic manifold andπ : G → G

H be a projection map. Then the symplectic form Ω, which is defined byg(X, JY) = Ω(X,Y) gives us a closed 2-form ω = (π∗Ω)e on G, where e is the identityon the Lie algebra G. Then a central extension G := G ⊕ Rξ of G can be defined by

[X,Y] = [X,Y]G − ω(X,Y)ξ, [ξ, X] = 0, ∀X,Y ∈ G, (6)

where ω = dξ. Also a global tensor ϕ of the type (1, 1) is defined by ϕξ = 0 andϕ(X) = JX. By using these facts and Theorem 6.1 in [5] we obtain the followingresult.

Theorem 5.1. Up to isomorphisms, All and ones five-dimensional homogeneousSasakian spaces (A4(λ, µ), ξ, g, ϕ) which are obtained by the contactization of (A4(λ, µ),J, gλ,µ) with λ = 2µ can be described as follows,(a) The Riemannian metric g can be written as g(X0, X0) = ±1, g(Xi, Xi) = 1.(b) The non-zero Lie brackets are

[X1, X2] = λX3 + X0, [X4, X1] = µX1, [X4, X2] = µX2, [X4, X3] = 2µX3 − X0. (7)

(c) The tensor field ϕ is given by ϕX1 = X2, ϕX3 = −X4, and ϕξ = 0, where ξ = X0.In all the above cases (a), (b) and (c), X0, X1, . . . , X4 is an pseudo-orthonormal basisof the Lie algebraA4(λ, µ) of A4(λ, µ).

[1] M. Aghasi and M. Nasehi, On the geometrical properties of solvable Lie groups, Adv. Geom. 15(2015) 507-517.

[2] D.V. Alekseevsky, Contact homogeneous spaces, Funct. Anal. Appl, 24 (1990) 324-325.[3] W. Batat, P. M. Gadea and J. A. Oubina, Homogeneous Riemannain structures on some solvable

extensions of the Heisenberg group, Acta Math. Hungar. (2012), 24 pages.[4] J. Berndt, F. Tricceri and L. Vanhecke, Generalized Heisenberg Groups and Damek-Ricci

Harmonic Spaces, Lecture Notes in Math., Vol. 1598, Springer-Verlag 1995.[5] G. Calvaruso, Symplectic, complex and Kähler structures on four-Dimensional Generalized

Symmetric Spaces, Differential Geom. Appl. 29 (2011) 758-769.[6] G, Calvaruso, Harmonicity of vector fields on four-dimensional generalized symmetric spaces

Cent. Eur. J. Math., 10 ( 2012) 411-425.[7] P. Petersen and P. Wylie, Rigidity of gradient Ricci Solitons, arxiv:0710.3174 (2009).

M. Nasehi,Department of Mathematical Sciences,Isfahan University of Technology,Isfahan, 84156-83111, Irane-mail: [email protected]

148



January 20–21, 2016

On the BSE-property of abstract Segal algebras

M. Nemati

Abstract

For a commutative semisimple Banach algebraA which is an ideal in its second dual we give a necessaryand sufficient conditions for an essential abstract Segal algebra in A to be a BSE-algebra. We also showthat elements of a large class of abstract Segal algebras in the Fourier algebra A(G) of a locally compactgroup G are BSE-algebra if and only if they have ∆-weak bounded approximate identities.

2010 Mathematics subject classification: Primary 46J10, 22D15; Secondary 43A30.Keywords and phrases: Abstract Segal algebra, commutative Banach algebra, Fourier algebra, locallycompact group.

1. Introduction

Let A be a commutative Banach algebra without order and let ∆(A) be the carrierspace of A. Recall that a bounded continuous function σ on ∆(A) is called a BS E-function if there exists a constant C > 0 such that for any elements φ1, ..., φn of ∆(A)and c1, ..., cn ∈ C the inequality∣∣∣∣∣∣∣∣

n∑j=1

c jσ(φ j)

∣∣∣∣∣∣∣∣ ≤ C

∥∥∥∥∥∥∥∥n∑

j=1

c jφ j

∥∥∥∥∥∥∥∥A∗holds, where A∗ denotes the dual space of A. Let CBS E(∆(A)) denote the set of allBSE-functions. For any σ ∈ CBS E(∆(A)) the infimum of all such C is denoted by∥σ∥BS E .

A multiplier onA is a bounded linear operator onA satisfying T (ab) = aT (b) forall a, b ∈ A. The set of all multipliers onA is denoted by M(A) which is a unital andcommutative Banach algebra. For each T ∈ M(A) there exists a unique continuousfunction T on ∆(A) such that T (a)(φ) = T (φ)a(φ) for all a ∈ A and φ ∈ ∆(A); see forexample [8, Theorem 1.2.2]. The algebra A is called a BSE-algebra (or to have theBSE-property) if

CBS E(∆(A)) = M(A).

This class of Banach algebras was first introduced and studied by Takahasi and Hatori[13] and then described by several authors for various kinds of Banach algebras; see

149

M. Nemati

[6, 7, 9]. Examples of BSE-algebras are the group algebra L1(G) of a locally compactabelian group G, the Fourier algebra A(G) of a locally compact amenable group G,all commutative C∗-algebras, the disk algebra, and the Hardy algebra on the open unitdisk.

A bounded net (eγ)γ in A is called a ∆-weak bounded approximate identity if itsatisfies φ(eγ) → 1 for all φ ∈ ∆(A). Such approximate identities were studied in [5].Note that ∆-weak bounded approximate identities are important to decide whether acommutative Banach algebra is a BSE-algebra or not. For example, it was shown in[4] that a Segal algebra S (G) on a locally compact abelian group G is a BSE-algebraif and only if it has a ∆-weak bounded approximate identity.

In this work, for a commutative semisimple Banach algebraA which is an ideal inits second dual we give a necessary and sufficient conditions for an essential abstractSegal algebra inA to be a BSE-algebra. Finally, we show that a large classe of abstractSegal algebras in the Fourier algebra A(G) of a locally compact group G are BSE-algebra if and only if they have ∆-weak bounded approximate identities.

2. BSE-property of abstract Segal algebras

Recall from Burnham [1] that a Banach algebra B is an abstract Segal algebra of aBanach algebraA if

(1) B is a dense left ideal inA;(2) there exists M > 0 such that ∥b∥A ≤ M∥b∥B for each b ∈ B;(3) there exists C > 0 such that ∥ab∥B ≤ C∥a∥A∥b∥B for each a, b ∈ B.Endow ∆(A) and ∆(B) with the Gelfand topology and recall from [1], Theorem

2.1, that the map φ 7→ φ|B is a homeomorphism from ∆(A) onto ∆(B). We saythat B is essential if ⟨AB⟩ is ∥ · ∥B-dense in B, where ⟨AB⟩ is the linear span ofAB = ab : a ∈ A, b ∈ B.

Theorem 2.1. Let A be a semisimple commutative Banach algebra which is an idealin its second dual A∗∗. Suppose that B is an essential abstract Segal algebra in A.Then the following statements are equivalent.

(i) B is a BSE-algebra.(ii) B = A andA is a BSE-algebra.

Proof. Suppose that B is a BSE-algebra. Then B has a ∆-weak bounded approximateidentity, say (gγ)γ. It is clear that (gγ)γ is also a ∆-weak bounded approximate identityfor A. By [7, Theorem 3.1] A is a BSE-algebra and has a bounded approximateidentity, say (eα)α. Since B is essential, (eα)α is also a bounded approximate identityfor B in A. In fact, for each b ∈ B and ε > 0, there is c =

∑ni=1 aibi with 1 ≤ i ≤ n,

ai ∈ A and bi ∈ B such that ∥b − c∥B ≤ ε. Thus for each α we have

∥eαb − b∥B ≤ (1 + K)ε +Cn∑

i=1

∥eαai − ai∥A∥bi∥B,

150


where K = sup ∥eα∥A. This shows that

∥eαb − b∥B → 0

for all b ∈ B. Thus, B = BA by Cohen’s factorization theorem. Now, let b ∈ B and(bn) be a bounded sequence in B. Then b = ca for some c ∈ B and a ∈ A. Since thesequence (bn) is also bounded inA andA is an ideal in its second dual, it follows thatthe operator ρa : A → A defined by ρa(a′) = aa′, (a′ ∈ A) is weakly compact by [2].Therefore, there exists a subsequence (bnk ) of (bn) such that (ρa(bnk )) is convergent tosome a′ in the weak topology of A. Now, we observe that f · c ∈ A∗ for all f ∈ B∗,where ( f · c)(a) = f (ca) for all a ∈ A. This shows that the sequence (ρca(bnk )) isconvergent to ca′ in the weak topology of B. It follows that the operator ρb : B → B isweakly compact which implies that B is an ideal in its second dual again by [2]. SinceB is semisimple, [7, Theorem 3.1] implies that B has a bounded approximate identity.ThusA = B by [1, Theorem 1.2 ]. That (ii) implies (i) is trivial.

Example 2.2. Let G be a compact abelian group and let S (G) be a Segal algebra onG; see [10] for more details. In this case it is well known that L1(G) is an ideal inits second dual. Moreover, recall that any Segal algebra is an essential abstract segalalgebra in L1(G). Therefore, above theorem implies that S (G) is a BSE-algebra ifand only if S (G) = L1(G), since L1(G) is a BSE-algebra; see [11]. This result wasoriginally obtained in [6].

Example 2.3. Let G be a locally compact group and let A(G) be the Fourier algebraof G, introduced by Eymard [3]. It was shown in [7, Theorem 5.1] that A(G) is aBSE-algebra if and only if G is amenable. Moreover, it is well known that A(G) is anideal in its second dual if and only if G is discrete. Therefore, by Theorem 2.1 if G isdiscrete, then the essential abstract Segal algebra S A(G) in A(G) is a BSE-algebra ifand only if S A(G) = A(G) and G is amenable.

Let G be a locally compact group and let Lr(G) be the Lebesgue Lr-space of G,where 1 ≤ r < ∞. Then

S Ar(G) := Lr(G) ∩ A(G)

with the norm ||| f ||| = ∥ f ∥r + ∥ f ∥A(G) and the pointwise product is an abstract Segalalgebra in A(G).

Corollary 2.4. Let G be a discrete group and let 1 ≤ r ≤ 2. Then S Ar(G) is a BSE-algebra if and only if G is finite

Proof. First note that S Ar(G) = lr(G) and the norms ∥ · ∥r and ||| · ||| on Ar(G) areequivalent. In fact, l2(G) ⊆ δe ∗ l2(G) ⊆ A(G), where δe is the point mass at the identityelement e of G. So, if 1 ≤ r ≤ 2, then lr(G) ⊆ l2(G) and

lr(G) ⊆ lr(G) ∩ l2(G) ⊆ S Ar(G) ⊆ lr(G).

Moreovere, it is clear that lr(G) has an approximate identity and consequently it isan essential abstract Segal algebra in A(G). Therefore, if S Ar(G) = lr(G) is a BSE-algebra, then A(G) = lr(G) by Example 2.3. Thus, A(G) = l2(G), is a reflexive predual

151

M. Nemati

of a W∗-algebra. This implies, as is known, that A(G) is finite dimensional, thus G isfinite.

For a locally compact group G, we recall that A(G) is always an ideal in the Fourier-Stieltjes algebra, B(G) and note that M(A(G)) = B(G) when G is amenable. Thespectrum of A(G) can be canonically identified with G. More precisely, the mapx 7→ φx, where φx(u) = u(x) for all u ∈ A(G), is a homeomorphism from G onto∆(A(G)).

Theorem 2.5. Let G be a locally compact group and let S A(G) be an abstract Segalalgebra in A(G) such that B(G) ⊆ M(S A(G)). Then S A(G) is a BSE-algebra if andonly if S A(G) has a ∆-weak bounded approximate identity.

Proof. Suppose that S A(G) is a BSE-algebra. Then S A(G) has a ∆-weak boundedapproximate identity by [13, Corollary 5 ].

Conversely, suppose that S A(G) has a ∆-weak bounded approximate identity, say(eγ)γ. Then M(S A(G)) ⊆ CBS E(∆(S A(G))) again by [13, Corollary 5]. Moreover, it isclear that (eγ)γ is also a ∆-weak bounded approximate identity for A(G). Consequently,we conclude that G is amenable by [7, Theorem 5.1]. Now, we need to show the reverseinclusion. Since S A(G) is an abstract Segal algebra in A(G), there exists M > 0such that ∥u∥A(G) ≤ M∥u∥S A(G) for all u ∈ S A(G). Thus, for any x1, ..., xn ∈ G andc1, ..., cn ∈ C, ∥∥∥∥∥∥∥∥

n∑j=1

c jφx j

∥∥∥∥∥∥∥∥S A(G)∗

≤ M

∥∥∥∥∥∥∥∥n∑

j=1

c jφx j

∥∥∥∥∥∥∥∥A(G)∗

.

This implies that

CBS E(∆(S A(G))) ⊆ CBS E(∆(A(G)))= B(G)⊆ M(S A(G)).

Hence, S A(G) is a BSE-algebra.

Corollary 2.6. Let G be a locally compact group and let S A(G) be an essential abstractSegal algebra in A(G). Then S A(G) is a BSE-algebra if and only if S A(G) has a ∆-weak bounded approximate identity.

Proof. Suppose that S A(G) has a ∆-weak bounded approximate identity. Then A(G)has a bounded approximate identity. Now, by a same argument during the proof ofLemma 2.1 we can show that S A(G) = A(G)S A(G). Consequently, uS A(G) ⊆ S A(G)for all u ∈ B(G) which implies that B(G) ⊆ M(S A(G)). Hence, S A(G) is a BSE-algebra by above theorem.

Example 2.7. (1) Let G be a locally compact group and let 1 ≤ r < ∞. Now, since∥u∥∞ ≤ ∥u∥B(G) for all u ∈ B(G), it follows that uLr(G) ⊆ Lr(G). This implies that

152


B(G) ⊆ M(S Ar(G)). Thus Ar(G) is a BSE-algebra if and only if it has a ∆-weakbounded approximate identity.

(2) Let S 0(G) be the Feichtinger’s Segal algebra in A(G). Then B(G) ⊆ M(S 0(G));see [12, Corollary 5.2]. Thus S 0(G) is a BSE-algebra if and only if it has a ∆-weakbounded approximate identity.

[1] J. T. Burnham, Closed ideals in subalgebras of Banach algebras. I, Proc. Amer. Math. Soc. 32(1972), 551–555.

[2] J. Duncan and S. A. R. Hosseiniun, The second dual of a Banach algebra, Proc. Roy. Soc.Edinburgh Sect. A 84 (1979), 309–325.

[3] P. Eymard, L’algébre de Fourier d’un groupe localement compact, Bull. Soc. Math. France, 92(1964), 181–236.

[4] J. Inoue and S.-E. Takahasi, Constructions of bounded weak approximate identities for Segalalgebras on LCA groups, Acta Sci. Math. (Szeged) 66 (2000), 257–271.

[5] C. A. Jones and C.D. Lahr, Weak and norm approximate identities are different, Pacific J. Math.72 (1977), 99–104.

[6] Z. Kamali andM. Lashkarizadeh Bami, Bochner-Schoenberg-Eberlein property for abstract Segalalgebras, Proc. Japan Acad. Ser. A Math. Sci. 89 (2013), 107–110.

[7] E. Kaniuth and A. Ülger, The Bochner-Schoenberg-Eberlein property for commutative Banachalgebras, especially Fourier and Fourier-Stieltjes algebras, Trans. Amer. Math. Soc. 362 (2010),4331–4356.

[8] R. Larsen, An introduction to the theory of multipliers, Springer-Verlag, New York, 1971.[9] M. Nemati and H. Javanshiri, The multiplier algebra and BSE-functions for certain product of

Banach algebras, preprin, arXiv:1509.00895, 2015.[10] H. Reiter, L1-algebras and Segal algebras, Lect. Notes Math. 231, Springer- Verlag, Berlin, 1971.[11] W. Rudin, Fourier analysis on groups, Interscience, New York, 1962.[12] E. Samei, N. Spronk and R. Stokke, Biflatness and pseudo-amenability of Segal algebras, Canad.

J. Math. 62 (2010), 845–869.[13] S.-E Takahasi and O. Hatori, Commutative Banach algebras which satisfy a Bochner-

Schoenberg- Eberlein type-theorem, Proc. Amer. Math. Soc. 110 (1990), 149–158.

M. Nemati,Department of Mathematical SciencesIsfahan University of TechnologyIsfahan 84156-83111, Irane-mail: [email protected]

153



January 20–21, 2016

Module extension Banach algebras and (σ, τ)-amenability ofBanach algebras

A. NiaziMotlagh

Abstract

In this paper we use homomorphisms to show that the Banach algebraA is (σ, τ)-amenable if and only ifFor each Banach algebra B and every injective homomorphism φ : A −→ B, H1

(σ,τ)(A,B∗φ) = 0

2010 Mathematics subject classification: Primary 46H25; Secondary 47B47.Keywords and phrases: (σ, τ)-derivation, (σ, τ)-inner derivation, (σ, τ)-amenability, (σ, τ)-contractibility,approximate identity, Banach algebra, Banach module.

1. Introduction

Let A be a Banach algebra and X be a Banach A-bimodule, that X is both a Banachspace and an algebraic A-bimodule, and the module operations (a, x) 7→ ax and(a, x) 7→ xa from A × X into X are (jointly) continuous. Then X∗ is also a BanachA-bimodule under the following module actions:

(a · f )(x) = f (xa),( f · a)(x) = f (ax),

a ∈ A, x ∈ X, f ∈ X∗.Let A be a Banach algebra. Given f ∈ A∗ and F ∈ A∗∗, then F f and f F are

defined inA∗ by the following formulae

F f (a) = F( f · a), f F(a) = F(a · f ) (a ∈ A)

Next, for F,G ∈ A∗∗, FG is defined inA∗∗ by the formulae

(FG)( f ) = F(G f )

this product is called first Arens product on A∗∗ and A∗∗ with the first Arens productis a Banach algebra.

Let A be a Banach algebra and X be a Banach A-bimodule. The Banach spaceX∗∗ is a BanachA∗∗-bimodule under following actions

F ·G = w∗ − limi

limj

aix j, G · F = w∗ − limj

limi

x jai

154

A. NiaziMotlagh

where F = w∗ − limi ai, G = w∗ − lim j x j.Suppose that φ : A → B is a Banach algebra homomorphism. The Banach algebra

B is considered as a BanachA- bimodule by the following module actions

a · b = φ(a)b, b · a = bφ(a) (a ∈ A, b ∈ B)

we denote Bφ the aboveA-bimodule.LetA be a Banach algebra and σ, τ be continuous homomorphisms onA. Suppose

that X is a Banach A-bimodule. A linear mapping d : A → X is called a (σ, τ)-derivation if

d(ab) = d(a)σ(b) + τ(a)d(b) (a, b ∈ A).

For example (i) Every ordinary derivation of an algebraA into anA-bimodule X is anidA-derivation, where idA is the identity mapping on the algebra A. (ii) Every pointderivation d : A → C at the character θ onA is a θ-derivation.

A linear mapping d : A −→ X is called (σ, τ)-inner derivation if there exists x ∈ Xsuch that d(a) = τ(a)x − xσ(a) (a ∈ A). See also [3–5, 7] and references therein.

We denote the set of continuous (σ, τ)-derivations from A into X by Z1(σ,τ)(A,X)

and the set of inner (σ, τ)-derivations by B1(σ,τ)(A,X). we define the space H1

(σ,τ)(A,X)as the quotient space Z1

(σ,τ)(A,X)/B1(σ,τ)(A,X). The space H1

(σ,τ)(A,X) is called thefirst (σ, τ)-cohomology group ofA with coefficients in X.

Let A be a Banach algebra and X be a Banach A-bimodule. Define A ⊕1 X byactions:

(a, x) + (b, y) = (a + b, x + y)a(b, x) = (ab, ax) , (b, x)a = (ba, xa)

(a, x)(b, y) = (ab, ay + xb)

for every a, b ∈ A and x, y ∈ X.It is clearA⊕1 X is a BanachA-bimodule with the following norm:

∥(a, x)∥ = ∥a∥ + ∥x∥

This Banach algebra is called module extension Banach algebra.We use some ideas and terminology of [2] to investigate (σ, τ)-amenability of

Banach algebras

2. (σ, τ)-amenability of Banach algebras

Let T : A → B be a continuous linear map between Banach algebras. Twocontinuous linear maps T ′ : B∗ → A∗ and T ′′ : A∗∗ → B∗∗ are known, that aredefined by the following formula(

T ′( f ))(a) = f

(T (a)

),

(T ′′(G)

)( f ) = G

(T ′( f )

)where a ∈ A, f ∈ B∗ and G ∈ A∗∗.

155

(σ, τ)-amenability of Banach algebras

Lemma 2.1. Let A be a Banach algebra, X be a Banach A-bimodule, and let σ andτ be two continuous homomorphisms on A. Suppose that D : A −→ X is (σ, τ)-derivation. Then D′′ : A∗∗ −→ X∗∗ is a (σ′′, τ′′)-derivation.

Proof. Let F,G ∈ A∗∗. Say F = w∗−limα aα,G = w∗−limβ bβ inA∗∗, where (aα), (bβ)are nets inA with ||aα|| ≤ ||F||, ||bβ|| ≤ ||G||. Then

D′′(FG) = D′′(w∗ − lim

αw∗ − lim

βaαbβ

)= w∗ − lim

αw∗ − lim

βD′′(aαbβ)

= w∗ − limα

w∗ − limβ

(τ(aα)D(bβ) + D(aα)σ(bβ)

)= τ′′(F)D′′(G) + D′′(F)σ′′(G)

and so D′′ is a (σ′′, τ′′)-derivation.

Now we are ready to state some equivalent conditions by (σ, τ)-amenability ofBanach algebras.

Theorem 2.2. Let σ and τ be two continuous homomorphisms on Banach algebraA.The following statements are equivalent:1. A is (σ, τ)-amenable.2. For each Banach algebra B and every homomorphism φ : A −→ B,

H1(σ,τ)(A,B∗φ) = 0

3. For each Banach algebra B and every injective homomorphism φ : A −→ B,H1

(σ,τ)(A,B∗φ) = 04. For each Banach algebra B and every injective homomorphism φ : A −→ B, if

d : A −→ Bφ∗ is a (σ, τ)-derivation satisfies

(d(a))(φ(b)) + (d(b))(φ(a)) = 0 (a, b ∈ A)

then d is (σ, τ)-inner derivation.

Proof. Clearly (1) ⇒ (2) ⇒ (3) ⇒ (4). It is sufficient to show that (4) ⇒ (1). Let Xbe a Banach A-bimodule and D : A −→ X∗ be a (σ, τ)-derivation. Set B = A ⊕1 Xand define injective homomorphism φ : A −→ B by φ(a) = (a, 0) and so we canassume that A is a subalgebra of B. Define d : A −→ B∗φ by d(a) = (0,D(a)). Themap d is (σ, τ)-derivation, since

d(ab) = (0,D(ab)) = (0,D(a)σ(b) + τ(a)D(b))= (0,D(a))(0, σ(b)) + (0, τ(a))(0,D(b))= d(a)φ(σ(b)) + φ(τ(a))d(b)= d(a) · σ(b) + τ(a) · d(b) (a, b ∈ A).

156

A. NiaziMotlagh

Since (d(a))(φ(b))+ (d(b))(φ(a)) = (0,D(a))((b, 0))+ (0,D(b))((a, 0)) = 0, we have(d(a))(φ(b)) + (d(b))(φ(a)) = 0.

It follows from our assumption that d is a (σ, τ)-inner derivation. Hence there aref ∈ A∗ and g ∈ X∗ such that

(0,D(a)) = d(a) = (σ(a), 0)( f , g) − ( f , g)(τ(a), 0)= (σ(a) f − f τ(a), σ(a)g − gτ(a))

Hence D is (σ, τ)-inner derivation. Moreover, σ(a) f = f τ(a). (a ∈ A)

Definition 2.3. LetA be a Banach algebra and σ be a continuous homomorphisms onA. The Banach algebraA is called approximately σ-contractible, if for each BanachA-bimodule X and σ-derivation D : A −→ X, there exists a bounded net xα ∈ X suchthat

D(a) = limα

(σ(a)xα − xασ(a)

)(a ∈ A)

In the following theorem we follow the structure of Proposition 2.8.59[1]

Theorem 2.4. LetA be a Banach algebra and σ be a bounded homomorphism onA.Then the following assertion are equivalent:

1. A is σ-amenable.

2. For everyA-module X, H1(σ,σ)(A,X∗∗) = 0

3. A is approximately σ-contractible.

Proof. (1) ⇒ (2) is trivially. (2) ⇒ (3): Let D : A −→ X be a σ-derivation from Ainto A-module X and let JX : X −→ X∗∗ be the canonical embedding, then for eacha, b ∈ A we have

D(ab) = (JX D)(ab) = JX(σ(a)D(b) + D(a)σ(b)

)= σ(a)D(b) + D(a)σ(b).

Thus D is a σ-derivation. Then by (2) there exists Λ ∈ X∗∗ such that D(a) =σ(a)Λ − Λσ(a) (a ∈ A). Set m = ||Λ||,U = X[m]. Then Λ ∈ JX(U)

w∗. Let

a1, a2, a3, . . . , an ∈ A, the V = Πnj=1

(σ(a j)U − Uσ(a j)

)is a convex subset of X(n)

and (D(a1),D(a2), . . . ,D(an)) ∈ Vweak. Thus for each finite subset F ofA, and ε > 0,

there exists x(F,ε) ∈ U such that

||D(a) − (σ(a)x(F,ε) − x(F,ε)σ(a))|| < ε (a ∈ F).

The family of such pairs (F, ε) is a directed if order ≤ given by

(F1, ε1) ≤ (F2, ε2)⇔ F1 ⊆ F2, ε1 ≤ ε2.

157


Thus we haveD(a) = lim

(F,ε)

(σ(a)x(F,ε) − x(F,ε)σ(a)

).

(c) ⇒ (a). Let D : A −→ X∗ be a σ-derivation. Then there exists a net (x′α) ⊆ X∗such that D(a) = limα

(σ(a)x′α − x′ασ(a)

)(a ∈ A). By passing to a subnet we may

assume that w∗ − lim x′α = x′ in X∗ and then D(a) = σ(a)x′ − x′σ(a). Thus A isσ-amenable.

Theorem 2.5. Let A be a Banach algebra and σ be a continuous homomorphism onA. IfA∗∗ is σ′′-amenable, thenA is σ-amenable.

Proof. Let X be a banach A-module, and D : A −→ X∗∗ be a σ-derivation. Then byLemma 2.1, D′′ : A∗∗ −→ X∗∗∗∗ is a σ′′-derivation. Since A∗∗ is σ′′-amenable, thenthere exists x(4) ∈ X∗∗∗∗ such that D′′(a′′) = σ′′(a′′)x(4) − x(4)σ′′(a′′), (a′′ ∈ A∗∗).We have X∗∗∗∗ = X∗∗ ⊕ (X∗)⊥ (as A∗∗-modules). Let P : X∗∗∗∗ −→ X∗∗ be the naturalprojection. Then for each a ∈ A, we have D(a) = σ(a)P(x(4)) − P(x(4))σ(a), and soD ∈ N1

(σ,σ)(A,X∗∗). Thus by above theorem,A is σ-amenable.

Proposition 2.6. LetA be a Banach algebra and σ be a continuous homomorphism onA. Then A is a σ-amenable Banach algebra if and only if for every Banach algebraB and every injective homomorphism φ : A −→ B, H1

(σ,σ)(A, B∗∗φ ) = 0.

Proof. One side is clear, so we prove the other side. Let X be a Banach A-bimoduleand D : A −→ X∗∗ be a σ-derivation. If ϕ : A −→ A⊕1 X is defined by φ(a) = (a, 0).It is easy to show that φ∗∗ : A∗∗ −→ (A⊕1 X)∗∗ the second transpose of φ is a Banachalgebra homomorphism and ((A⊕1 X)φ)∗∗ ≃ (A∗∗ ⊕1 X∗∗)φ∗∗ asA∗∗-bimodules. Then

H1(σ,σ)(A, (A∗∗ ⊕1 X∗∗)φ∗∗) = H1

(σ,σ)(A, ((A⊕1 X)φ)∗∗) = 0 (1).

Now we define D1 : A −→ A∗∗ ⊕1 X∗∗ by D1(a) = (0,D(a)). For a, b ∈ A wehave D1(ab) = D1(a)φ∗∗(b) + φ∗∗(a)D1(b). Thus D1 is a σ-derivation from A into(A∗∗ ⊕1 X∗∗)φ∗∗ . By (1), D1 is inner. Therefore there exists a′′ ∈ A∗∗, x′′ ∈ X∗∗ suchthat D1 = d(a′′,x′′), and by a similar proof in Theorem 2.2 we can show that D is inner.Then we have H1

(σ,σ)(A,X∗∗) = 0, and by Theorem 2.4,A is σ-amenable.

Proposition 2.7. Let A be a Banach algebra. Then A has a bounded approximateidentity if and only ifA is (id, 0) and (0, id)-amenable.

3. An example

We use Banach algebra introduced by Yong Zhang [10] to obtain necessarycondition for (σ, τ)-amenability of this Banach algebra.

It is easy to see that ℓ1 is a Banach algebra equipped with the following product

a · b = a(1)b (a, b ∈ ℓ1).

158

A. NiaziMotlagh

and ℓ1 has a left identity e defined by

e(n) =

1 i f n = 10 i f n , 1

The dual space (ℓ1)∗ = ℓ∞ is a ℓ1-bimodule via the ordinary actions as follows

a · f = f (a)e, f · a = a(1) f (a ∈ ℓ(S ), f ∈ ℓ∞)

where e is regarded as an element of ℓ∞.Next let σ : ℓ1 −→ ℓ1 be a bounded homomorphism. we have a(1)σ(b) = σ(a·b) =

σ(a)·σ(b) = σ(a)(1)σ(b) and so σ(b)(a(1)−σ(a)(1)) = 0 for all a, b ∈ N. Since σ , 0,we have (

σ(a))(1) = a(1) (a ∈ ℓ1) (1)

In [5] has been shown that ℓ1 is (σ, τ)-weak amenable for all homomorphisms σ, τbut for some homomorphisms σ and τ it is not (σ, τ)-amenable. In the following weprove if the Banach algebra ℓ1 is (σ, τ)-amenable, then τ(a) = a(1)c where c(1) = 1.

Let B = ℓ1 by product a • b = a(2)b. Then B is a Banach algebra and for eachbounded homomorphism ψ : B −→ B we have

(ψ(a)

)(2) = a(2). Let a ∈ ℓ1 define

a′ ∈ ℓ1 by a′ =(a(2), a(1), a(3), · · ·

).Define mapping φ : ℓ1 −→ B by φ(a) = a′. It

is clear φ is a homomorphism. Consider the Banach ℓ1-bimodule Bφ under actionsa b = φ(a) • b = a′ • b = a′(2)b = a(1)b and b a = b • φ(a) = b • a′ = b(2)a′ foreach a ∈ ℓ1, b ∈ Bφ. Let D : ℓ1 −→ B∗φ be a bounded (σ, τ)-derivation. We have(

D(a · b))(c) = D(a)σ(b)(c) + τ(a)D(b)(c)

a(1)D(b)(c) = D(a)(σ(b) c) + D(b)(c τ(a))a(1)D(b)(c) = b(1)D(a)(c) + c(2)D(b)(τ(a))

for all a, b ∈ ℓ1 and c ∈ Bφ.By taking a = b we obtain D(a)(τ(a)) = 0. Also by taking c ∈ Bφ such that

c(2) = 0 we can conclude a(1)D(b) = b(1)D(a).By (σ, τ)-amenability of ℓ1 there exists f ∈ B∗φ such that D = D f is a (σ, τ)-inner

derivation. We have

a(1)D f (b) = b(1)D f (a)a(1) f (b(1)c − c(2)τ(b)) = b(1) f (a(1)c − c(2)τ(a))

for all a, b ∈ ℓ1 and c ∈ Bφ.Then f (b(1)c(2)τ(a) − a(1)c(2)τ(b)) = 0. Since f ∈ B∗φ is arbitrary, immediately

is conclude a(1)τ(b) = b(1)τ(a). By taking b = e we have τ(a) = a(1)τ(e), whereτ(e)(1) = 1.Then we have the following theorem

Theorem 3.1. Let σ, τ be two continuous homomorphisms on ℓ1. If ℓ1 is a (σ, τ)-amenable then τ(a) = a(1)c where c(1) = 1.

159


[1] H. G. Dales, Banach algebra and Automatic continuity, Oxford university Press, 2001.[2] M. E. Gorgi, Homomorphisms, Amenability and weak amenability of Banach algebras,

arXiv:math.FA/0610105 v1.[3] M. Mirzavaziri, M. S. Moslehian, σ-derivations in Banach algebras, Bull. Iranian Math. Soc. 32,

no. 1, (2006) 65–78[4] M. S. Moslehian, Approximate (σ − τ)-contractibility, Nonlinear Funct. Anal. Appl., 11 , no. 5,

(2006) 805–813.[5] M. S. Moslehian and A. N. Motlagh, Some notes on (σ, τ)-amenability of Banach algebras,

STUDIA UNIV. " BABES-BOLYAI", MATHEMATICA,volume LIII, Number 3, September 2008.[6] M. Mirzavaziri andM. S. Moslehian, Automatic continuity ofσ-derivations in C∗-algebras, Proc.

Amer. Math. Soc., 11, no. 5, (2006) 805–813.[7] M. S. Moslehian, Approximately vanishing of topological cohomology groups, J. Math. Anal.

Appl. 318 (2006) 758–771.[8] T. W. Palmer, Banach algebras and the general theory of ∗-algebras, Vol. I. Algebras and Banach

algebras, Encyclopedia of Mathematics and its Applications 49, Cambridge University Press,Cambridge, 1994.

[9] J.P Pier, Amenable Banach Algebra, Pitman research notes in mathematical series 172, Longe-man, Essex, 1988.

[10] Yong Zhang, Weak Amenability of a Class of Banach Algebras, Canada. Math. Bull. Vol. 44(4),(2001) 504-508.

A. NiaziMotlagh,Department of Mathematics,Faculty of basic sciences,University of Bojnord,P. O. Box 1339, Bojnord, Iran.,

e-mail: [email protected] and [email protected]

160



January 20–21, 2016

Derivation on Hilbert H∗-modules

M. Niknam∗ and E. Keyhani

Abstract

We define a derivation from a Hilbert H∗-module X into Hilbert H∗ -module X as a linear map δ such thatδ([x, y]z) = [δ(x), y]z + [x, δ(y)]z + [x, y]δ(z) for all x, y, z ∈ X. For such a derivation the main results statethat under conditions δ|AnnX (σ(δ)) is continuous in which σ(δ) is a separating space for δ and AnnX(σ(δ)) isa modular annihilator for σ(δ).

2010 Mathematics subject classification: Primary 47L30; 17A36..Keywords and phrases: Derivation, Hilbert H∗-module, Separating space, Modular annihilator..

1. Introduction

Definition 1.1. [1] An H∗-algebra is a B-algebra which satisfies the following furtherconditions:(i) The underlying Banach space of A is a Hilbert space (of arbitrary dimension);(ii) For each x ∈ A there is an element in A, denoted by x∗ and called an adjoint of x,such that for all y, z in A we have both < xy, z >=< y, x∗z > and < xy, z >=< x, zy∗ >(here,as throughout this paper, the symbol < x, y > stands for the Hilbert space innerproduct of x and y).

The following facts about adjoints in an H∗-algebra are obvious:(1) if x∗ is an adjoint of x, then x is an adjoint of x∗,(2) if x∗ and y∗ are adjoints of x and y respectively, then λx∗ + µy∗ is an adjoint ofλx + µy, and y∗x∗ is an adjoint of xy,(3) every element of the form xx∗ or x∗x is self-adjoint, that is,is an adjoint of itself.

Example 1.2. The Hilbert space l2 = an : an ∈ C,∑

n

|an|2 < ∞ is a H∗-algebra,

where for each an and bn in l2, anbn = anbn and an∗ = an.

Recall that A0 = a ∈ A : aA = 0 = a ∈ A : Aa = 0 is called the annihilatorideal of A. A proper H∗-algebra is a H∗-algebra with zero annihilator ideal.The trace class τ(A) of A is defined by the set ab : a, b ∈ A. It is known that∗ speaker

161

M. Niknam and E.Keyhani

τ(A) is an ideal of A which is a Banach ∗-algebra under a suitable norm τA(.). Thenorm τA is related to the given norm ∥.∥ on A by τA(a∗a) = ∥a∥2 (a ∈ A)and ∥a∥ ≤ τA(a) for each a ∈ τ(A) (see[9]). The trace functional tr on τ(A) isdefined by tr(ab) =< a, b∗ >=< b, a∗ >= tr(ba) for each a, b ∈ A, in particulartr(aa∗) = tr(a∗a) = ∥a∥2 for all a ∈ A.The notion of Hilbert H∗-module is introduced by Saworotnow [8] under the nameof generalized Hilbert space. It has been studied by Smith [11], Cabrera, Martinez,Rodriguez [4] and others.

Definition 1.3. [3] Let A be a H∗-algebra. A Hilbert H∗-module is a left module Eover A with a mapping [., .] : E × E → τ(A), which satisfies the following conditions:(i) [αx, y] = α[x, y],(ii) [x + y, z] = [x, z] + [y, z],(iii) [ax, y] = a[x, y],(iv) [x, y]∗ = [y, x],(v) for each nonzero element x in E there is a nonzero element a in A such that[x, x] = a∗a,(vi) E is a Hilbert space with the inner product (x, y) = tr([x|y]),for each α ∈ C, x, y, z ∈ E, a ∈ A. For example every H∗-algebra A is a Hilbert A-module whenever we define [x, y] = xy∗. We say Hilbert A-module E is full, if the linearspace generated the set [x, y] : x, y ∈ E is τA dense in τ(A). In particular, we shallfrequently use the following three immediate consequences of the above definition:

∥x∥2 = tr([x, x]) = τ([x, x]) ∀x ∈ E,

∥[x, y]∥ ≤ τ([x, y]) ≤ ∥x∥∥y∥ ∀x, y ∈ E,

∥ax∥ ≤ ∥a∥∥x∥ ∀a ∈ A, x ∈ E.

Recall that a derivation of an algebra A is linear mapping δ from A into itself suchthat δ(ab) = δ(a)b + aδ(b) (a, b ∈ A). (for further results see [5, 6]).

A linear mapping δ : X → X is called a derivation on Hilbert H∗-module X, if itsatisfies

δ([x, y]z) = [δ(x), y]z + [x, δ(y)]z + [x, y]δ(z), (1)

for every x, y, z ∈ X.

Example 1.4. Let A be a Hilbert H∗-module over unitary algebra A with [x, y] =xy∗ (x, y ∈ A). Then every ∗-derivation on A in the above sense is a derivation on analgebra A. In fact,

δ([x, y]z) = [δ(x), y]z + [x, δ(y)]z + [x, y]δ(z),

thus

δ(xy∗z) = δ(x)y∗z + x(δ(y))∗z + xy∗δ(z). (2)

162


If x = y = z = 1, then we have

δ(1) = δ(1) + (δ(1))∗ + δ(1).

So δ(1) = 3δ(1), implies δ(1) = 0. In view of (2), with z = 1 we derive

δ(xy∗) = δ(x)y∗ + xδ(y∗),

and (2), becomes

δ(xy∗z) = δ(xy∗)z + xy∗δ(z). (3)

Since A is unitary, the Cohen factorization’s theorem [12] satisfies, hence using (3)with u = xy∗ and v = z, we deduce

δ(uv) = δ(u)v + uδ(v).

Conversely, if δ is derivation on an algebra A, then δ is aderivation in the sence (1).Indeed,

δ(xy∗z) = δ(xy∗)z + (xy∗)δ(z)= (δ(x)y∗ + xδ(y∗))z + xy∗δ(z)= δ(x)y∗z + xδ(y∗)z + xy∗δ(z).

The separating space σ(S ) of a linear operator S from a Banach space X into aBanach space Y is defined as belows:

σ(S ) = y ∈ Y : there is a sequence (xn) in X with xn → 0 and S xn → y.

The closed graph theorem implies that S is continuous if and only if σ(S ) = 0 (see[7, 10]).

In 1974, W. G. Bade and P. C. Jr. Curtis [2] proved that if A is a Banach algebra,D be a module derivation from A into a A-module M and I(D) be the continuiuty idealof D and I(D) has a bounded approximate identity, then the restriction of D on I iscontinuous.

2. Main results

Lemma 2.1. Let X be a proper Hilbert H∗-module. Assume that δ : X → X is aderivation, then σ(δ) is a normed close submodule of X.

Corollary 2.2. Let X be simple Hilbert H∗-module and σ(δ) , X, then δ is continuous.

Given an element x in X. Consider the map Tx : X → X by Tx(y) = [x, y]x.Obviously, Tx is a bounded anti-linear operator. The following formula is alwayssatisfied TXTyTx = TT x(y) (x, y ∈ X).

For each submodule M of X, we define its modular annihilator, AnnX(M), as theset x ∈ X : Tx(y) = [x, y]x = 0 ∀y ∈ M. From this immediately, we conclude that[x, y]x ∈ AnnX(M) (x ∈ AnnX(M), y ∈ X).

163

M. Niknam and E.Keyhani

Lemma 2.3. Let X be a Hilbert H∗-module with this property that δ(xn) is convergentfor each sequence xn that is convergent to zero. Given an element x in X, then the mapTxδ : X → X is a continuous linear mapping if and only if σ(δ) ⊆ ker(Tx).

Theorem 2.4. Let X be a Hilbert H∗-module and δ : X → X be a derivation with thisproperty that δ(xn) is convergent for every sequence xn converges to zero. Assumethat AnnX(σ(δ)) is a normed closed submodule of X such that Ly(y) ∈ AnnX(σ(δ))implies that y ∈ AnnX(σ(δ)) and

< AnnX(σ(δ)), AnnX(σ(δ)) > σ(δ) =< AnnX(σ(δ)), σ(δ) > AnnX(σ(δ)) = 0. (4)

Then δ|AnnX(σ(δ)) : AnnX(σ(δ))→ X is continuous.

Acknowledgement

The authors would like to thanks the anonymous referee for his/her comments thathave been implemented in the final version of the paper.

[1] W. Ambrose, Structure theorems for a special class of Banach algebras. Transactions of theAmerican Mathematical Society 57, (1945), 364-386.

[2] W.G. Bade and P. C. Curtis, Prime ideals and automatic continuity problems for Banach algebras,Journal of Functional Analysis. 29. 1(1978), 88–103.

[3] D. Bakic, B. Guljas, Operators on Hilbert H∗-modules. Journal of Operator Theory 46, (2001),123-137.

[4] M. Cabrera, J. Martinez and A. Rodriguez, Hilbert modules revisited: Orthonormal bases andHilbert-Schmidt operators. Glasgow Mathematical Journal 37, (1995), 45-54.

[5] M. Hassani and E. Keyhani, On the superstability of a special derivation, Journal of Linear andTopological Algebra, 2014, 15–22.

[6] A. Hosseini, M. Hassani, A. Niknam, Generalized σ-derivation on Banach algebras, Bulletin ofthe Iranian Mathematical Society, 37 no. 4(2011), 81–94.

[7] J.R. Ringrose, Automatic continuity of derivations of operator algebras, J. London Math. Soc. 5(1972), no. 2. 432–438.

[8] P. P. Saworotnow, A generalized Hilbert space. Duke Mathematical Journal 35, (1968), 191-197[9] P. P. Saworotnow, J. C. Friedell, Trace-class for an arbitrary H∗-algebra. Proceedings of the

American Mathematical Society 26, (1970), 95-100.[10] A.M. Sinclair, Automatic continuity of linear operators, Cambridge University Press, vol 21,

1976.[11] J. F. Smith, The structure of Hilbert modules. Journal of the London Mathematical Society 8,

(1974), 741-749[12] W. Zelazko, Banach algebras. Elsevier Pub. Co. 1973.

M. Niknam,Department of Mathematics, Hamedan Branch,Islamic Azad University,Hamedan, Iran.e-mail: [email protected]

164


E. Keyhani,Department of Mathematics, Mashhad Branch,Islamic Azad University,Mashhad, Iran.e-mail: [email protected].

165



January 20–21, 2016

Crossed Product of Triple Operator Spaces

H. Nikpey

Abstract

Let V be an TRO-operator space and iso(V) be the group of all completely isometric bijective linearmappings on V . Let G act on V via a continuous group homomorphism α : G → iso(V). We define theoperator space crossed product V ⋊tr

α G and show that for a C∗-algebra with its canonical operator spacestructure, this coincide with the C∗-algebra crossed product.

2000 Mathematics subject classification: Primary 46L07; Secondary 47L65.Keywords and phrases: Crossed Product, Triple Operator Spaces (TRO).

1. Introduction

Let B(H) denote the algebra of bounded linear operators on a Hilbert space H. Aconcrete operator space is a closed subspace of B(H). These include C∗-algebras andtheir closed subspaces. There is also an abstract formulation of operator space due toZ-J. Ruan [3].

Given operator spaces V ⊆ B(H) and W ⊆ B(K), a linear mapping φ : V → W,lifts to the matrix level as φn : Mn(V) → Mn(W) defined by φn(T ) = [φ(Ti, j)] forT = [Ti, j] ∈ Mn(V). The completely bounded norm of φ is

∥φ∥cb = sup∥φn∥ : n ∈ N,

where the norms ∥φn∥ come from the canonical embeddings Mn(V) ⊆ B(Hn) andMn(W) ⊆ B(Kn) and form an increasing sequence

∥φ∥ ≤ ∥φ2∥ ≤ · · · ≤ ∥φn∥ ≤ · · · ≤ ∥φ∥cb.

The map φ is completely bounded (respectively, a completely contraction or a com-plete isometry) if ∥φ∥cb < ∞ (respectively, ∥φ∥cb ≤ 1 or each φn is an isometry). Theoperator spaces V and W are completely isometric if there is a surjective completeisometry φ : V → W. In this case we write V W. For more detail see [2]

We recall that a Triple (Ternary) Ring of Operators or TRO is a closed linearsubspace V of B(K,H) (or of a C∗-algebra) satisfying VV∗V ⊆ V .

The main objective of this paper is to define crossed products of TRO withtopological groups acting on them. The basic idea is to make the underlying operator

166

Please supply the list of authors

space into a C∗-algebra using an auxiliary product and finding an appropriate operatorsubspace C∗-algebra crossed product as the candidate to play the role of operator spacecrossed product.

2. Crossed Product of TRO

A C∗-dynamical system consists of a locally compact group G acting by automor-phisms on a C∗-algebra A. Here is a more formal definition.

Definition 2.1. A C∗-dynamical system is a triple (A,G, α) consisting of a C∗-algebraA, a locally compact, Hausdorff group G and a strongly continuous homomorphismα : G → Aut(A).

A pair of representations (ρ,U) with ρ : A → B(H) a (non-degenerate) ∗-representation of A and U : G → U(H) a unitary representation of G is called α-covariant if

ρ(α(g)(a)) = U(g)ρ(a)U∗(g)

for each a ∈ A and g ∈ G. For f ∈ Cc(G, A), define

ρ ⋊ U( f ) =∫

Gρ( f (g))U(g)dg,

and put∥ f ∥ = sup∥ρ ⋊ U( f )∥

where the supremum is taken over all α-covariant pairs (ρ,U). This is a norm onCc(G, A), called the universal norm. The completion of Cc(G, A) with respect to thisnorm is called the full crossed product of A by G, denoted by A ⋊α G (see [4] for moredetails).

Let V be a TRO in B(K,H). Let G be a locally compact group with a continuoushomomorphism α : G → iso(V). Let

δ : V → B(K2,K1)

be a TRO morphism i.e. δ(T1T ∗2T3) = δ(T1)δ(T2)∗δ(T3) for each T1,T2 and T3 ∈ Vand

U =[U1 00 U2

]: G → B(K1 ⊕ K2)

be a unitary representation such that

δ(α(g)(T )) = U1(g)δ(T )U2(g)∗.

In this case, (δ,U) is called an α-covariant representation. As B(K2,K1) is the 1-2corner of B(K1 ⊕ K2), we may define

δ : V → B(K2,K1) → B(K1 ⊕ K2)

167

Crossed Product of Triple Operator Spaces

by

T → δ(T ) →[0 δ(T )0 0

].

For f ∈ Cc(G,V), put

∥ f ∥V⋊trαG = sup∥

∫Gδ( f (g))U2(g)dg∥

where the supremum is taken over all α-covariant representations (δ,U). Define V⋊trαG

to be the completion of Cc(G,V) in this norm.

Proposition 2.2. Let A be a C∗-algebra and α : G → Aut(A) be a continuous action.Then

A ⋊α G A ⋊trα G.

Let V be a right A-C∗-module. Let L(V) =[KA(V) V

V∗ A

]be linking C∗-algebra

generated by V , for more detail see [1, Section 8]. Thus V is TRO. Let

α =

[α1 α2α∗2 α3

]: G → Aut(L(V))

be a dymaical system. We can find relationship between V ⋊trα G and L(V) ⋊α G

[1] D. Blecher and C. Le Merdy, Operator Algebras and their Modules: An Operator SpaceApproach,

[2] E. Effros and Z.J. Ruan, Operator Spaces, London Math. Soc. Monographs, New Series 23,Oxford University Press, New York, 2000.

[3] Z.-J. Ruan, Subspaces of C∗-algebras, J. Funct. Anal. 76 (1988), 217-230.[4] D. P. Williams, Crossed Products of C∗-algebras, Mathematical surveys and monographs 134,

American Mathematical Society, Providence, 2007.

H. Nikpey,Department of Mathematics,Shahid Rajaee Teacher Training University of Tehran, Iran.e-mail: [email protected]

168



January 20–21, 2016

Structure of some Beurling algebras on left coset spaces

B. Olfatian gillan

Abstract

Let G be a locally compact group and H be a compact subgroup of G. We consider the homogeneousspace G/H equipped with a strongly quasi-invariant Radon measure µ which arises from a rho-function.We introduce a new Beurling algebra L1

ω(G/H), where ω is a weight function on the homogeneous spaceG/H. Then we study the structure of this Beurling algebra. As example, we show that L1

ω(G/H) may beidentified with a quotient space of L1

ωq(G) where q : G → G/H is the canonical quotint map. Also, weprove that L1

ω(G/H) can be considered as a Banach subalgebra of L1ωq(G).

2010 Mathematics subject classification: Primary 43A15; Secondary 43A85, 46B25.Keywords and phrases: Locally compact topological group, Homogeneous space, Beurling algebra,Mackey-Bruhat formula, Lau algebra.

1. Introduction

When G is a locally compact group with fixed left Haar measure λ, a continuousfunction ω : G → [1,∞) is called a weight function on G, if

ω(xy) ≤ ω(x)ω(y) (x, y ∈ G).

The space of all measurable functions f such that fω ∈ L1(G) is called a Beurlingalgebra which is denoted by L1

ω(G). The Beurling algebra L1ω(G) under the norm ∥.∥1,ω

defined by∥ f ∥1,ω = ∥ fω∥1 ( f ∈ L1

ω(G)),

is a Banach algebra. The dual space of L1ω(G) is L∞ω (G); the Lebesgue space as defined

in [7], which formed by all complex-valued measurable function φ on G such thatφ/ω ∈ L∞ω (G). In fact, the continuous linear functionals on L1

ω(G) are precisely thoseof the form

⟨φ, f ⟩ =∫

Gf (x)φ(x)dx ( f ∈ L1

ω(G), φ ∈ L∞ω (G)). (1)

The Lebesgue space L∞ω (G) with the product defined by

φ ·ω ψ = φψ/ω (φ, ψ ∈ L∞ω (G)),

169

B. Olfatian gillan

and the norm ∥ · ∥∞,ω defined by

∥φ∥∞,ω = ∥φ/ω∥∞ (φ ∈ L∞ω (G)),

and the complex conjugation as involution is a commutative C∗-algebraLet H be a compact subgroup of locally compact topological group G. In general,

we know that the space G/H consisting of all left cosets of H in G is not a group.But in [2], it is shown that G/H is a homogeneous space which can be equipped witha strongly quasi-invariant Radon measure µ which arises from a rho-function ρ. By[7, Remark 8.2.3], for any given rho-function ρ for the pair (G,H), there is a stronglyquasi-invariant Radon measure µ on G/H such that for each f ∈ L1(G) we have∫

G/H

∫H

fρ

(xξ)dξdµ(xH) =∫

Gf (x)dx,

which it is called the Mackey-Bruhat formula. In [3], assuming that H is a compactsubgroup of G, the operator T1 : L1(G) → L1(G/H, µ) is defined by T1 f (xH) =∫

Hf (xξ)ρ(xξ) dξ for almost all xH ∈ G/H. Then it is shown that L1(G/H) with the

multiplication given byφ ∗ ψ(xH) = T1(φρ ∗ ψρ)

is a Banach algebra with a right approximate identity, where φ, ψ ∈ L1(G/H),φρ = ρ(φ q), ψρ = ρ(ψ q) and q : G → G/H is the canonical quotint map.

In this paper, the basic aim is to extend the definition of the weight functionω on a homogeneous space G/H . Then we introduce and study a new Beurlingalgebra L1

ω(G/H) as a Banach subalgebra of L1(G/H). We show that L1ω(G/H)

may be identified with a quotient space of L1ωq(G) where q : G → G/H is the

canonical quotint map. Also, we prove that L1ω(G/H) can be considered as a Banach

subalgebra of L1ωq(G) which this help us show that the Beurling algebra L1

ω(G/H)alweys possesses a right approximate unit. At the end, we show that L1

ω(G/H) is alsoa Lau algebra.

2. Definition of a Beurling algebra on a homogeneouse space

Definition 2.1. Let G be a locally compact Hausdorff topological group and H be acompact subgroup of G. A real-valued function ω on the homogeneous space G/H issaid to be a weight function if it has the following properties:(1) ω(xH) ≥ 1 x ∈ G,(2) ω(xyH) ≤ ω(xH)ω(yH) x, y ∈ G,(3) ω is measurable and locally bounded.

It is straightforward to see that if ω is a weight function on G/H, then ω q is aweight function on G. Also, L1

ωq(G) is a Banach subalgebra of L1(G) under the norm∥ · ∥1,ωq in which ∥ f ∥1,ωq =

∫G | f (x)|ω q(x)dx for any f ∈ L1

ωq(G) (cf. [7]). Wedenote the space of all measurable functions f such that fω ∈ L1(G/H) by L1

ω(G/H).We can easily check that ∥ · ∥1,ω defined by ||φ||1,ω =

∫G/H |φ(xH)|ω(xH)dµ(xH) where

170

Structure of some Beurling algebras on left coset spaces

φ ∈ L1ω(G/H), is a norm on the L1

ω(G/H). So, L1ω(G/H) is a normed subspace of

L1(G/H).From now on, we assume that ω is a weight function on the homogeneous space

G/H and T1 is the mapping introduced in section 1.

Lemma 2.2. The following properties for all f ∈ L1ωq(G) and φ ∈ L1

ω(G/H) hold:(1) ∥T1 f ∥1,ω ≤ ∥ f ∥1,ωq,(2) ∥φ∥1,ω = ∥φρ∥1,ωq.

By using Lemma 2.2, we can write

∥φ∥1,ω ≤ inf∥ f ∥1,ωq : f ∈ L1ωq(G), T1 f = φ.

Since for all φ ∈ L1ω(G/H) we have ∥φ∥1,ω = ∥φρ∥1,ωq, so

∥φ∥1,ω = inf∥ f ∥1,ωq : f ∈ L1ωq(G), T1 f = φ,

thus we can say that T1 induces an isometry isomorphism between L1ωq(G)/ ker T1 and

L1ω(G/H). Hence L1

ω(G/H) may be identified with a quotient space of L1ωq(G),

Proposition 2.3. (L1ω(G/H), ||.||1,ω) is a Banach subalgebra of L1(G/H).

Proof. Let L1ωq(G : H) be the set of φρ;φρ ∈ L1

ωq(G), φ ∈ L1(G/H). ThenL1ωq(G : H) is a closed subalgebra of L1(G) (cf. [8]). Also, by using Lemma 2.2

we can say that the restriction of T1 from L1ωq(G : H) onto L1

ω(G/H) is an isometryisomorphism. So L1

ω(G/H) is a Banach algebra under the norm ||.||1,ω.

We call the Banach algebra (L1ω(G/H), ||.||1,ω) a Beurling algebra on G/H.

Corollary 2.4. The Beurling algebra L1ω(G/H) can be considered as a Banach subal-

gebra of L1ωq(G).

Proposition 2.5. The Beurling algebra L1ω(G/H) possesses a right approximate unit.

For each a ∈ G and φ ∈ L1ω(G/H), we define the left (resp. right) translation La

(resp. Ra) as Laφ = T1(Laφρ) (resp. Raφ = T1(Raφρ)). In the sequel, we present themore facts about L1

ω(G/H).

Proposition 2.6. Suppose a ∈ G and φ ∈ L1ω(G/H). Then we have the following

assertions:(1) ||Laφ||1,ω ≤ ω(aH)∥φ∥1,ω,(2) ||Raφ||1,ω ≤ ω(aH)∥φ∥1,ω.

Note that the above proposition implies that L1ω(G/H) is invariant with respect to

the left and right translations.

Proposition 2.7. The mapping a → Laφ(a → Raφ) from G into L1ω(G/H) is a

continuous function, where a ∈ G and φ ∈ L1ω(G/H).

171

B. Olfatian gillan

[1] J.W. Baker and A. Rejali, On the Arens regularity of weighted convolution algebras, J. London.Math. Soc. 40 (1989), 535–546.

[2] G.B. Folland, A Course in Abstract Harmonic Analysis, CRC Press, Boca Raton, 1995.[3] R.A. Kamyabi-Gol and N. Tavallaei, Convolution and homogeneous spaces, Bull. Iranian Math.

Soc. 35 (2009), no. 1, 129–146.[4] E. Kaniuth, Weak spectral synthesis in Fourier algebras of coset spaces, Studia mathematica. 197

(2010), no. 3, 229–246.[5] A.T. Lau, Analysis on a class of Banach algebras with applicatoins on locally compact groups

and semigroups, Fund. math. 118 (1983), 161–175.[6] K. Parthasarathy and N. Shravan Kumar, Fourier algebras on homogeneous spaces, Bulletin

des sciences mathematiques, 135 (2011), no. 2, 187–205.[7] H. Reiter and J.D. Stegeman, Classical Harmonic Analysis and Locally Compact Groups, Oxford

Science Publications, Clarendon Press, New York, 2000.[8] N. Tavallaei, M. Ramezanpour and B. Olfatian-Gillan, Structral transition between Lp(G) and

Lp(G/H), Banach J. Math. Anal. 9 (2015), no. 3, 194–205.

B. Olfatian gillan,Department of Mathematics and Computer Science,University of Damghan,City of Damghan, Iran.e-mail: [email protected]

172



January 20–21, 2016

Topics in the concept of amenability modulo an ideal of Banachalgebras

H. Rahimi

Abstract

The main purpose of this paper is to investigate amenability of semigroups and present a characterizationof amenability of semigroup algebras restoring to Johnson’s theorem for semigroups. We relate this to anew notion of amenability of Banach algebras modulo an ideal and consider some different variations ofamenability modulo an ideal of Banach algebras.

2010 Mathematics subject classification: Primary 99X99, Secondary 99Y99.Keywords and phrases: Amenability modulo an ideal, amenability, group congruence, semigroup algebra.

1. Introduction

The theory of amenable Banach algebras begins with B.E. Johnson in 1972 [8]. Theterminology comes from [8]: a locally compact group G is amenable if and onlyif the group algebra L1(G) is amenable. Over the years, many different variationsof amenability have been introduced, among which one can refer to [11] that is agood survey of various types of amenability. Different notions of amenability ofthe semigroup algebra l1(S ) have been widely studied in the recent years. For mostof these notions of amenability, it is not known in general when 1(S ) possess thesenotions of amenability. In fact, for a discrete semigroup S , the necessary and sufficientconditions for amenability of the semigroup algebra l1(S ) is rather more complicated.It is shown that for an E-unitary inverse semigroup S with finitely many idempotents,l1(S ) is amenable if and only if each maximal group of S is amenable [3] (see also [4],where the E-unitary restriction is dropped). In general, l1(S ) is amenable if and only ifS is "built up from amenable groups" [5]. For a commutative semigroup S , this meansthat S should be a finite semilattice of abelian groups [7]. Thus, it is natural to searchfor an alternative way to make the Johnson’s idea work for semigroups.

In this paper, we briefly review the concept of amenability modulo an ideal, somecharacterizations of this concept and its applications to the investigation of amenabilityof semigroup algebras. For more details we refer the reader to [1, 9, 10].

173

H. Rahimi

2. Amenability modulo an ideal and its relation to amenability

Throughout this section, A is a Banach algebra, X is a Banach A-bimodule and I isa closed ideal of A.

Definition 2.1. a) A Banach algebra A is amenable modulo I if for every Banach A-bimodule E such that I · E = E · I = 0, and every derivation D from A into E∗ there isϕ ∈ E∗ such that D = adϕ on the set theoretic difference A\I := a ∈ A : a < I.

b) An element M ∈ ( AI ⊗A)∗∗ is a virtual diagonal modulo I if

a · π∗∗AI

M − a = 0 (a ∈ A, a = a + I)

a · M − M · a = 0 (a ∈ A \ I)

c) A bounded net (mα)α ⊂ AI ⊗A is a approximate diagonal modulo I if

a · π AImα − a→ 0 (a ∈ A, a = a + I)

a · mα − mα · a→ 0 (a ∈ A \ I)

d) A Banach algebra A is contractible modulo I if for every Banach A-bimodule Xsuch that I · X = X · I = 0, every bounded derivation D from A into X is an innerderivation on the set theoretic difference A\I := a ∈ A : a < I.

Theorem 2.2. Let I be a closed ideal of A.a) If A/I is amenable and I2 = I then A is amenable modulo I.b) If A is amenable modulo I then A/I is amenable.c) If A is amenable modulo I and I is amenable, then A is amenable.d) A is amenable modulo I if and only if there is an approximate diagonal modulo

I, if and only if there is a virtual diagonal modulo I.e) If A

I is contractible and I2 = I then A is contractible modulo I.e) If A is contractible modulo I then A

I is contractible.f) If A is contractible modulo I and I is contractible then A is contractible.g) If A is contractible modulo I then A

I be an unital.

Remark 2.3. (i) The concept of amenability modulo an ideal is a generalization of theconcept of amenability.

(ii) As we mentioned in Theorem 2.2, amenability modulo a close ideal I impliesamenability of A

I but the converse is not true.(iii) The generalized notions of amenability modulo an ideal as; weak amenability

modulo an ideal, approximate amenability modulo an ideal, contractibility modulo anideal and some others are introduced which one of them is a generalization of the itscorresponding classic notion.

3. Amenability of Semigroup algebras modulo an ideal

Throughout this section, S is a discrete semigroup and ρ is a group congruence onS .

174

Topics in the concept of amenability modulo an ideal

Theorem 3.1. Let ρ be a group congruence on S such that Kerρ is central, then S isamenable if and only if S/ρ is amenable.

Theorem 3.2. Let S be a semigroup, ρ be a group congruence on S such that Ker(ρ)is central and Iρ has an approximate identity. Then S is amenable if and only if l1(S )is amenable modulo Iρ .

Theorem 3.3. Let S be an E-inversive E-semigroup with commuting idempotents.Then S is amenable if and only if S/σ is amenable.

Theorem 3.4. Let S be an E-inversive E-semigroup with commuting idempotents.Then S is amenable if and only if l1(S ) is amenable modulo Iσ.

A semigroup S is called eventually inverse if every element of S has some powerthat is regular and E(S ) is a semilattice. If S is eventually inverse semigroup, then therelation σ = (s, t) : es = et for some e ∈ E(S ) is the least group congruence on S[6].

By an argument similar to that of Theorem 3.3, for H = E(S ), we have thefollowing result.

Theorem 3.5. Let S be an eventually inverse semigroup. Then S is amenable if andonly if S/σ is amenable.

Corollary 3.6. Let S be an eventually inverse semigroup. Then S is amenable if andonly if l1(S ) is amenable module Iσ.

Theorem 3.7. If S is either(i) E-inverse E-semigroup with commutative idempotents, or(ii) eventually inverse semigroup with commutative idempotents,then l1(S ) is contractible modulo Iσ if and only if S

σis finite.

[1] M. Amini and H. Rahimi, Amenability of semigroups and their algebras modulo a groupcongruence, Acta Mathematica Hungarica, Vol 144, Issue 2 , (2014), pp 407-415

[2] M. Day, Amenable semigroups, Illinois J. Math. 1, 509-544 (1957).[3] J. Duncan , I. Namioka, Amenability of inverse semigroups and their semigroup algebras, Proc.

Royal Soc. Edinburgh Sect. A 80, 309-321 (1978).[4] J. Duncan and A. L. T. Paterson, Amenability for discrete convolution semigroup algebras, Math.

Scandinavica 66, 141-146 (1990).[5] G. K. Dales, A. T.-M. Lau and D. Strauss, Banach Algebras on Semigroups and their Compacti-

fications, Memoirs American Mathematical Society, Vol. 205, No. 966, American MathematicalSociety, Providence (2010).

[6] G. M. S. Gomes, Group congruences on eventually inverse semigroups, Portugaliae Mathematica49.4, 417-428 (1992).

[7] N. Groenbaek, Amenability of discrete convolution algebras, the commutative case, Pacific J.Math. 143, 243-249 (1990)

[8] B. E. Johnson, Cohomology in Banach Algebras, Memoirs American Mathematical Society, Vol.127, American Mathematical Society, Providence (1972).

175

H. Rahimi

[9] H. Rahimi and E. Tahmasebi, Amenability and Contractibility modulo an ideal of Banachalgebras, Abstract and Applied Analysis, (2014), 514761.

[10] H. Rahimi and E. Tahmasebi, A note on amenability modulo an ideal of unitial Banach algebras,J. Mathematical Extension, Vol. 9, No. 1, 13-21, (2015).

[11] V. Runde, Lectures on Amenability. Lecture notes in mathematics, 1774, Springer Verlag, (2002).

H. Rahimi,Department of Mathematics, Faculty of Science,Central Tehran Branch, Islamic Azad University,Tehran, Iran.e-mail: [email protected]

176



January 20–21, 2016

On n-weak amenability of Lau product of Banach algebras

M. Ramezanpour

AbstractThe present note deals with the concept of n-weak amenability for Lau product of Banach algebras. Itextends some results on the n-weak amenability of the unitization of a Banach algebra to the Lau productof Banach algebras. In particular, it improves some results of n-weak amenability for Lau product ofBanach algebras and answers some questions on this topic.

2010 Mathematics subject classification: Primary 46H05, Secondary: 47B47, 43A20.Keywords and phrases: Banach algebra, n-weak amenability .

1. Introduction

Let A be a Banach algebra, and X be a Banach A-bimodule. Then the dual space X∗ ofX becomes a dual Banach A-bimodule with the module actions defined by

( f a)(x) = f (ax), (a f )(x) = f (xa),

for all a ∈ A, x ∈ X and f ∈ X∗. Similarly the n-th dual X(n) of X is a Banach A-bimodule. In particular, A(n) is a Banach A-bimodule. A derivation from A into X is alinear mapping D : A→ X satisfying

D(ab) = D(a)b + aD(b) (a, b ∈ A).

If x ∈ X then dx : A→ X defined by dx(a) = ax − xa is a derivation. A derivation D isinner if there is x ∈ X such that D = dx.

A Banach algebra A is called n-weakly amenable, for an integer n ≥ 0, if everycontinuous derivation from A into A(n) be inner, where A(0) = A. The algebra A is saidto be weakly amenable if it is 1-weakly amenable. The concept of weak amenabilitywas first introduced by Bade, Curtis and Dales in [? ] for commutative Banachalgebras, and was extended to the noncommutative case by Johnson [? ]. Dales,Ghahramani and Gronbeak [? ] initiated and intensively developed the study of n-weak amenability of Banach algebras.

Let A and B be Banach algebras and θ ∈ σ(B), the set of all nonzero multiplicativelinear functionals on B. The θ-Lau product A×θ B is a Banach algebra which is definedas the vector space A × B equipped with the algebra multiplication

(a1, b1)(a2, b2) = (a1a2 + θ(b2)a1 + θ(b1)a2, b1b2) (a1, a2 ∈ A, b1, b2 ∈ B),

177

M. Ramezanpour

and the norm ∥(a, b)∥ = ∥a∥ + ∥b∥. This type of product was introduced by Lau [?] for certain class of Banach algebras known as Lau algebras and was extended bySangani Monfared [? ] for the arbitrary Banach algebras. The unitization A♯ of A canbe regarded as the ι-Lau product A ×ι C where ι ∈ σ(C) is the identity map.

Some aspects of A×θ B such as many basic properties, some notions of amenabilityand some homological properties are investigated by many authors; see for example[? ? ] and [? ? ]. In particular, Ebrahimi Vishki and Khoddami [? ], studied then-weak amenability of A×θ B. They gave some necessary and sufficient conditions forn-weak amenability of A ×θ B. In the case when A is unital, they shown that, A ×θ Bis n-weakly amenable if and only if both A and B are n-weakly amenable. It was alsoleft as an open question whether this result holds for the case when A has a boundedapproximate identity. In this note, we improve some results of n-weak amenability forA ×θ B and partially answer to this question.

2. Main results

Throughout the section n is assumed to be a non-negative integer, A and B beBanach algebras and θ be an element of σ(B).

We start with the following theorems which extend some well known results on(2n + 1)-weak amenability of A♯ to A ×θ B.

Theorem 2.1. Suppose that A has a bounded approximate identity. Then A ×θ B is(2n + 1)-weakly amenable if and only if both A and B are (2n + 1)-weakly amenable.

Theorem 2.2. Suppose that A is weakly amenable. Then A ×θ B is (2n + 1)-weaklyamenable if and only if both A and B are (2n + 1)-weakly amenable.

The next result, for B = C and θ = ι, has already been proved by Dales,Ghahramani and Gronbaek in [? , Proposition 1.4].

Theorem 2.3. Suppose that A and B are commutative. Then A ×θ B is (2n + 1)-weaklyamenable if and only if both A and B are (2n + 1)-weakly amenable.

We recall from [? ] that B is called left (resp. right) θ-amenable if every continuousderivation from B into X∗ is inner, for every Banach B-bimodule X with b · x = θ(b)x(resp. x · b = θ(b)x); (b ∈ B, x ∈ X). This notion of amenability is a generalization ofthe left amenability of a class of Banach algebras studied by Lau in [? ], known as Laualgebras. Example of left (resp. right) θ-amenable Banach algebras include amenableBanach algebras and the Fourier algebra A(G) for a locally compact group G.

The next proposition, extends the related results on (2n)-weak amenability of A♯.

Proposition 2.4. Let A and B are (2n)-weakly amenable and ⟨A2⟩ be dense in A. ThenA ×θ B is (2n)-weakly amenable if one of the following statements holds.(i) B is weakly amenable.(ii) B is left (resp. right) θ-amenable.

For the converse of Proposition 2.4 we have the following.

178

On n-weak amenability of Lau product of Banach algebras

Proposition 2.5. Suppose that A ×θ B is (2n)-weakly amenable and n ≥ 1. Then A andB are (2n)-weakly amenable if one of the following statements holds.(i) A has a bounded approximate identity.(ii) B is (2)-weakly amenable.

From Proposition 2.4 and Proposition 2.5 we obtain also the following result whichextends [? , Theorem 3.1].

Theorem 2.6. If A has a bounded approximate identity, B is either weakly amenableor left (right) θ-amenable and n ≥ 1. Then A×θ B is (2n)-weakly amenable if and onlyif both A and B are (2n)-weakly amenable.

It was shown in [? , Theorem 3.1] that if A is unital then the (n)-weak amenabilityof A ×θ B is equivalent to the (n)-weak amenablility of both A and B. It was left asan open question for the case when A has a bounded approximate identity; see [? ,Remark 3.1]. If we combine Theorems 2.1, 2.2, 2.3 and 2.6, we have the followingtheorem which answers this question.

Theorem 2.7. Suppose that A has a bounded approximate identity, B is either weaklyamenable or left (right) θ-amenable and n ≥ 1. Then A×θ B is (n)-weakly amenable ifand only if both A and B are (n)-weakly amenable. If A is unital then the equivalenceis also true for n = 0.

As a consequence of above theorem, with A = C and θ ∈ σ(B), we have the nextresult.

Corollary 2.8. The θ-Lau product C ×θ B is (n)-weakly amenable if and only if B is(n)-weakly amenable.

179



January 20–21, 2016

Some Banach algebra properties of A ×T B

N. Razi∗ and A. Pourabbas

Abstract

Let T be a homomorphism from a Banach algebra B to a Banach algebra A. The Cartesian product spaceA × B with T -Lau multiplication and ℓ1-norm becomes a new Banach algebra A ×T B. We investigatethe notions such as approximate amenability, pseudo amenability, ϕ-pseudo amenability, ϕ-biflatnessand ϕ-biprojectivity for Banach algebra A ×T B. We also present an example to show that approximateamenability of A and B is not stable for A ×T B. Finally we characterize the double centralizer algebra ofA ×T B and present an application of this characterization. We also characterize Hochschild cohomologyfor the Banach algebra A ×T B.

2010 Mathematics subject classification: Primary: 46M10. Secondary: 46H25, 46M18..Keywords and phrases: approximately amenable, pseudo amenable, ϕ-pseudo amenable, ϕ-biflat, ϕ-biprojective, double centralizer algebra, Hochschild cohomology.

1. Introduction

Let A and B be Banach algebras and let T : B → A be an algebra homomorphism.Then we consider A × B with the following product

(a, b) ×T (c, d) = (ac + T (b)c + aT (d), bd) ((a, b), (c, d) ∈ A × B).

The Cartesian product space A × B with this product is denoted by A ×T B. Let A andB be Banach algebras and let ∥T∥ ≤ 1. Then we consider A ×T B with the followingnorm

∥(a, b)∥ = ∥a∥ + ∥b∥ ((a, b) ∈ A ×T B).

We note that A ×T B is a Banach algebra with this norm.Suppose that T : B → A is an algebra homomorphism with ∥T∥ ≤ 1 and A is a

commutative Banach algebra. Then Bhatt and Dabshi [1] have studied the properties,such as Gelfand space, Arens regularity and amenability of A×T B. Moreover supposethat A is unital with unit element e and ψ0 : B→ C is a multiplicative linear functionalon B. If we define T : B → A by T (b) = ψ0(b)e, then the product ×T coincideswith the Lau product [5]. The group algebra L1(G), the measure algebra M(G), theFourier algebra A(G) of a locally compact group G and the Fourier-Stieltjes algebra of∗ speaker

180

N. Razi and A. Pourabbas

a topological group are the examples of Lau algebra [5]. Lau product was extended bySangani Monfared for the general case [10]. The purpose of this paper is to determinesome homological properties of A ×T B for every Banach algebras A and B, such asapproximate amenability, pseudo amenability, φ-pseudo amenability, φ-biflatness andφ-biprojectivity. After all we characterize the double centralizer algebra of A×T B andwe will give an application of this characterization. We also characterize Hochschildcohomology for the Banach algebra A ×T B.

2. Second section

We recall some basic definition of the homological properties. A Banach algebraA is called biprojective if πA : A⊗A −→ A has a bounded right inverse whichis an A−bimodule map. A Banach algebra A is called biflat if the adjoint mapπA∗ : A∗ −→ (A⊗A)∗ of πA has a bounded left inverse which is an A−bimodule map.

Here the product morphism πA : A⊗A −→ A for a Banach algebra A is defined byπA(a ⊗ b) = ab. It is clear that πA is an A−bimodule map.

A Banach algebra A is called approximately amenable if for every A-bimoduleX, any derivation D : A → X∗ is approximately inner. A Banach algebra A has anapproximate identity if there is a net ηα ⊆ A such that lim ηαa − a = lim aηα − a = 0for all a ∈ A. A Banach algebra A has a weak approximate identity if there is a netηα ⊆ A such that lim f (ηαa − a) = lim f (aηα − a) = 0 for all a ∈ A and f ∈ A∗.

The notion of pseudo amenability of Banach algebras was introduced by Ghahra-mani and Zhang in [3]. A Banach algebra A is said to be pseudo amenable if thereexists a net ρα ⊆ A⊗A, such that aρα − ραa→ 0 and πA(ρα)a→ a for all a ∈ A [3].

The notion of character pseudo amenability was introduced by Nasr-Isfahani andNemati in [7]. Let A be a Banach algebra and let ϕ ∈ ∆(A), where ∆(A) is the characterspace of A. We say that A is ϕ-pseudo amenable if it has a (right)ϕ-approximatediagonal, that is, there exists a not necessarily bounded net (mα) ⊆ A⊗A such that

ϕ(πA(mα))→ 1 and ∥a · mα − ϕ(a)mα∥ → 0

for all a ∈ A [7].Let A be a Banach algebra and let ϕ ∈ ∆(A). Then A is called ϕ-biprojective if there

exists a bounded A-bimodule morphism ρ : A → A⊗A such that ϕ πA ρ(a) = ϕ(a)for every a ∈ A. A Banach algebra A is called ϕ-biflat if there exists a bounded A-bimodule morphism ρA : A → (A⊗A)∗∗ such that ϕ πA

∗∗ ρ(a) = ϕ(a) for everya ∈ A, where ϕ is a unique extension of ϕ on A∗∗ defined by ϕ(F) = F(ϕ) for everyF ∈ A∗∗. It is clear that this extension remains to be a character on A∗∗ [8]. Recallthat A∗, the dual of a Banach algebra A, is a Banach A−bimodule with the moduleoperations defined by

⟨ f · a, b⟩ = ⟨ f , ab⟩ , ⟨a · f , b⟩ = ⟨ f , ba⟩ , ( f ∈ A∗, a, b ∈ A).

Let A be a Banach algebra. The second dual A∗∗ of A with Arens products and which are defined by ⟨mn, f ⟩ = ⟨m, n · f ⟩, where ⟨n · f , a⟩ = ⟨n, f · a⟩ , and

181

Short title of the paper for running head

similarly ⟨mn, g⟩ = ⟨n, f · m⟩, where ⟨ f · m, a⟩ = ⟨m, a · f ⟩ for every a ∈ A, f ∈ A∗

and m, n ∈ A∗∗, becomes a Banach algebra.A Banach A-bimodule X is called neo-unital if for every x ∈ X there exist a, a′ ∈ A

and y, y′ ∈ X such that ay = x = y′a′.The dual space (A ×T B)∗ is identified with A∗ × B∗ via

⟨( f , g), (a, b)⟩ = f (a) + g(b), (a ∈ A, b ∈ B, f ∈ A∗, g ∈ B∗)

for more details see[6, Theorem 1.10.13]. Also the dual space (A ×T B)∗ is (A ×T B)-bimodule with the module operations defined by

(a, b) · ( f , g) = (a · f + T (b) · f , (a · f ) T + b · g)

( f , g) · (a, b) = ( f · a + f · T (b), ( f · a) T + g · b).

Moreover A ×T B is a Banach A-bimodule under the module actions a′ · (a, b) =(a′, 0) ×T (a, b) and (a, b) · a′ = (a, b) ×T (a′, 0), for all a, a′ ∈ A, b ∈ B. SimilarlyA ×T B is a Banach B-bimodule.

3. Third Section

Theorem 3.1. Let A, B be Banach algebras and let T : B → A be an algebrahomomorphism with ∥T∥ ≤ 1. Then

(i) If A ×T B is (ϕ, ϕ T )-biflat for ϕ ∈ (A), then A is ϕ-biflat.(ii) If A ×T B is (0, ψ)-biflat for ψ ∈ (B), then B is ψ-biflat.

Theorem 3.2. Let A and B be Banach algebras, let T : B → A be an algebrahomomorphism with ∥ T ∥≤ 1 and let ϕ ∈ (A) and ψ ∈ (B). Then

(i) A ×T B is (ϕ, ϕ T )-biprojective if and only if A is ϕ-biprojective.(ii) A ×T B is (0, ψ)-biprojective if and only if B is ψ-biprojective.

Lemma 3.3. Let A and B be Banach algebras and let T : B → A be an algebrahomomorphism with ∥T∥ ≤ 1. Then

(i) If (eα, ηα) is (bounded) weakly approximate identity for A×T B, then eα+T (ηα)and ηα are (bounded) weakly approximate identities for A and B, respectively.

(ii) If eα and ηβ are (bounded) weakly approximate identities for A and B,respectively, then (eα − T (ηβ), ηβ) is (bounded) weakly approximate identityfor A ×T B.

Theorem 3.4. Let A and B be Banach algebras and let A ×T B be approximatelyamenable for an algebra homomorphism T : B → A with ∥T∥ ≤ 1. Then A and Bare approximately amenable.

Example 3.5. Let l1 denote the well-known space of complex sequences and let K(l1)be the space of all compact operators on l1. It is well known that the Banach algebraK(l1) is amenable. We renorm K(l1) with the family of equivalent norm ∥ · ∥k such

182


that its bounded left approximate identity will be the constant 1 and its bounded rightapproximate identity will be k + 1.

Now if we consider c0-direct-sum A = ⊕∞k=1(K(l1), ∥ · ∥k), then A has a bounded leftapproximate identity but no bounded right approximate identity, see [? , page 3931]for more details. Now by choosing T = 0 we have A ⊕ Aop = A ×T Aop. Note that Aand Aop, the opposite algebra, are boundedly approximately amenable [? , Theorem3.1], but sine A ⊕ Aop has no bounded approximate identity, it is not approximatelyamenable [? , Theorem 4.1].

Theorem 3.6. Let A and B be Banach algebras and let A ×T B be pseudo-amenablefor an algebra homomorphism T : B → A with ∥T∥ ≤ 1. Then A and B are pseudo-amenable.

Theorem 3.7. Let A and B be Banach algebras and let T : B → A be an algebrahomomorphism with ∥T∥ ≤ 1. Then

(i) A ×T B is (ϕ, ϕ T )-pseudo amenable if and only if A is ϕ-pseudo amenable,(ii) A ×T B is (0, ψ)-pseudo amenable if and only if B is ψ-pseudo amenable,

where ϕ ∈ ∆(A) and ψ ∈ ∆(B).

If A is a Banach algebra, then the idealizer Q(A) of A in (A∗∗,) is defined by

Q(A) = f1 ∈ A∗∗ : x f1 and f1x ∈ A f or every x ∈ A,

see [9] for more details.

Proposition 3.8. Let A and B be Banach algebras and let T : B → A be an algebrahomomorphism with ∥T∥ ≤ 1. Then Q(A ×T B) Q(A) × Q(B).

Theorem 3.9. Let A and B be Banach algebras and let T : B → A be an algebrahomomorphism with ∥T∥ ≤ 1. If A ×T B has weakly bounded approximate identity,then M(A ×T B) M(A) × M(B).

Proof. Since A×T B has weakly bounded approximate identity, [9, Theorem 2] showsthat the map ϕ : Q(A ×T B) → M(A ×T B) defined by ϕ( f1, f2) = (L( f1, f2),R( f1, f2)) forevery ( f1, f2) ∈ Q(A ×T B) is onto. Set

KA×T B = ( f1, f2) ∈ A∗∗ ×T ∗∗ B∗∗ : (g1, g2)( f1, f2) = 0, ∀(g1, g2) ∈ (A∗∗ ×T ∗∗ B∗∗).

By [9, Theorem 1] ker ϕ = KA×T B ∩ Q(A ×T B). This implies that

Q(A ×T B)KA×T B ∩ Q(A ×T B)

M(A ×T B), (1)

by a similar argument, we have

Q(A)KA ∩ Q(A)

M(A) (2)

183


and

Q(B)KB ∩ Q(B)

M(B), (3)

where denotes the algebra isomorphism.Now we define ψ : KA×T B → KA ×KB by ψ( f1, f2) = ( f1 + T ∗∗( f2), f2). Clearly ψ is

an algebra isomorphism. Hence we have KA×T B KA × KB and by Proposition 3.8 wehave

Q(A ×T B) Q(A) × Q(B). (4)

Therefore

KA×T B ∩ Q(A ×T B) (KA × KB) ∩ (Q(A) × Q(B))=(KA ∩ Q(A)) × (KB ∩ Q(B)).

(5)

The equations (4) and (5) imply that

Q(A ×T B)KA×T B ∩ Q(A ×T B)

Q(A) × Q(B)

(KA ∩ Q(A)) × (KB ∩ Q(B))

Q(A)

KA ∩ Q(A)× Q(B)

KB ∩ Q(B).

(6)

Combining (1), (2) and (3) yields M(A ×T B) M(A) × M(B).

Theorem 3.10. Let A and B be Banach algebras and let T : B → A be an algebrahomomorphism with ∥T∥ ≤ 1. Suppose that A is approximately amenable, A hasa bounded approximate identity and B is amenable. Then A ×T B is approximatelyamenable.

We recall that the n-th Hochschild cohomology group Hn(A, X) is defined by thefollowing quotient,

Hn(A, X) =Zn(A, X)Bn(A, X)

,

Theorem 3.11. Let A and B be Banach algebras with bounded approximate identityand let T : B → A be an algebra homomorphism with ∥T∥ ≤ 1. Let E be an essentialA ×T B-bimodule. Then

H1(A ×T B, E∗) ≃ H1(A, E∗) ×H1(B, E∗),

where ≃ denotes the vector space isomorphism.We also recall that the n-th Hochschild cohomology group Hn(A, X) is defined by

the following quotient,

Hn(A, X) =Zn(A, X)Bn(A, X)

.

184


Proof. By [2, Theorem 2.9.53] we haveH1(A×T B, E∗) ≃ H1(M(A×T B), E∗), whereM(A ×T B) denotes the double centralizer algebra of A ×T B. But from [? , Theorem4.3] we have

M(A ×T B) M(A) ×M(B),

where denotes the algebra isomorphism. this implies that

H1(M(A ×T B), E∗) ≃ H1(M(A) ×M(B), E∗),

whereM(A) andM(B) denote the double centralizer algebra of A and B, respectively.Hence we have

H1(A ×T B, E∗) ≃ H1(M(A) ×M(B), E∗),

Using [4, Theorem 4] we obtain H1(M(A) × M(B), E∗) ≃ H1(M(A), E∗) ×H1(M(B), E∗), thus we have

H1(A ×T B, E∗) ≃ H1(M(A), E∗) ×H1(M(B), E∗).

Since E is an essential Banach A-bimodule and an essential Banach B-bimidule, wehave

H1(M(A), E∗) ≃ H1(A, E∗)

andH1(M(B), E∗) ≃ H1(B, E∗).

This completes the proof.

Corollary 3.12. Let A and B be Banach algebras with bounded approximate identityand let T : B → A be an algebra homomorphism with ∥T∥ ≤ 1. Let E be an essentialA ×T B-bimodule. Then

Hn(A ×T B, E∗) ≃ Hn(A, E∗) ×Hn(B, E∗)

for every n ≥ 1.

[1] S. J. Bhatt and P. A. Dabshi, Arens regularity and amenability of Lau product of Banach algebrasdefined by a Banach algebra morphism, Bull. Aust. Math. Soc. 87 (2013), 195-206.

[2] H. G. Dalse, Banach algebras and automatic continuity, London Mathematical Society Mono-graphs 24, Clarendon Press, Oxford, 2000.

[3] F. Ghahramani and Y. Zhang, pseudo-amenable and pseudo-contractible Banach algebras, Math.Proc. Camb. Phil. Soc. 124 (2007), 111-123.

[4] G. Hochschild, On the cohomology theory for associative algebras, Annals Math. 47 (1946), 568-579.

[5] A. T. Lau, Analysis on a class of Banach algebras with applications to harmonic analysis onlocally compact groups and semigroups, Fund. Math. 118 (3) (1983), 161-175.

[6] R. E. Megginson, An Introduction to Banach Space Theory, Springer, New York, (1998).[7] R. Nasr-Isfahani and M. Nemati, Cohomology characterization of character pseudo-amenability

Banach algebras, Bull. Aust. Math. Soc. 84 (2011), 229-237.

185


[8] A. Sahami, A. Pourabbas; On ϕ-biflat and ϕ-biprojective Banach algebras, Bull. Belgian Math.Soc. Simon Stevin, Vol. 20, 5 (2013), 789-801.

[9] K. Saito, A characterization of double centralizer algebras of Banach algebras, Sei. Rep. NiigataUniv. Ser A. 11 (1974), 5-11.

[10] M. Sangani Monfared, On certain products of Banach algebras with applications to harmonicanalysis, Studia Math. 178 (3) (2007), 277-294.

N. Razi,Faculty of Mathematics and Computer Science,Amirkabir University of Technology,424 Hafez Avenue, Tehran 15914, Irane-mail: [email protected]

A. Pourabbas,Faculty of Mathematics and Computer Science,Amirkabir University of Technology,424 Hafez Avenue, Tehran 15914, Irane-mail: [email protected]

186



January 20–21, 2016

Convolutions on the Haagerup tensor products of Fourieralgebras

M. Rostami∗ and N. Spronk

Abstract

We study the ranges of the maps of convolution u ⊗ v 7→ u ∗ v and a ‘twisted’ convolution u ⊗ v 7→ u ∗ v(u(s) = u(s−1)) and on the Haagerup tensor product of a Fourier algebra of a compact group A(G) withitself. In particular, we look into the problem of factoring these maps through projective and operatorprojective tensor products. We notice that (A(G), ∗) is an operator algebra and observe a set of spectralsynthesis.

2010 Mathematics subject classification: Primary 46L07, Secondary 43A10.Keywords and phrases: Fourier algebra, convolution, Haagerup tensor product.

1. Introduction

Let G be a compact group and A(G) be its Fourier algebra, in the sense of [5].In [7], questions of the following nature were addressed: what are the ranges of

convolution and ‘twisted’ convolution, when applied to A(G × G) = A(G)⊗A(G)(operator projective tensor product). The authors’ motivation was two-fold. First,these particular maps played a fundamental role in the famous result of B. Johnson([9]) that A(G) is sometimes non-amenable, and the authors were interested in seeinghow these techniques related to the completely bounded theory of Fourier algebras.Secondly, it was observed that ‘twisted’ convolution averages A(G×G) over left cosetsof the diagonal subgroup ∆ = (s, s) : s ∈ G, whereas convolution averages A(G ×G)over orbits of the group action (r, (s, t)) 7→ (sr−1, rt) : G × (G × G) → G × G. Thusthe images may be rightly regarded as Fourier algebras of certain homogeneous/orbitspaces of G×G in the sense of [6]. The homogeneous space G×G/∆ and the orbit spaceG ×G/∆ may be naturally identified with G. Thus we define Γ, Γ : C(G ×G)→ C(G)by

Γu(s) =∫

Gu(sr, r) dr and Γu(s) =

∫G

u(sr, r−1) dr.

It is easy to check that Γ(u ⊗ v) = u ∗ v and Γ(u ⊗ v) = u ∗ v.

∗ speaker

187

M. Rostami and N. Spronk

In [4], it was shown that the Haagerup tensor product A(G) ⊗h A(G) is a Banachalgebra. By [11, Thm. 2] this algebra has spectrum G × G. We shall note, below,that A(G) ⊗h A(G) is, in fact, semi-simple, and may thus be regarded as an algebra offunctions on G × G. Hence it is natural to ask whether we discover anything new ifwe apply the maps Γ and Γ to A(G) ⊗h A(G). While we gain no new spaces, we learninteresting comparisons between A(G) ⊗h A(G), A(G ×G) and C(G) ⊗h C(G).

2. Second section

2.1. Some basic results We let for each π in G, Trigπ = spans 7→ ⟨π(s)ξ| η⟩ : ξ, η ∈Hπ and Trig(G) =

⊕π∈G Trigπ. This has linear dual space Trig(G)′ =

∏π∈G B(Hπ)

via dual pairing⟨u, T ⟩ =

∑π∈G

dπTr(u(π)Tπ) (1)

where u(π) =∫

G u(s)π(s−1) ds. If T ∈ Trig(G)′, we let T be defined by ⟨u, T ⟩ = ⟨u,T ⟩in the duality (1). Here u(s) = u(s−1).

We will identify the left regular representation up to quasi-equivalence as

λ =⊕π∈G

π on H = ℓ2-⊕π∈G

Hπ.

It is standard that λ(G) is weak*-dense in Trig(G)′ in terms of the duality (1). The vonNeumann algebra generated by λ(G) is thus the operator space direct product

VN(G) = ℓ∞-⊕π∈G

B(Hπ).

The Fourier algebra is the predual of VN(G) via the dual pairing (1). Hence weobtain complete isometric identification

A(G) = ℓ1-⊕π∈G

dπS1dπ (2)

where S1d denotes the d × d-matrices with trace norm; i.e. for u in A(G) we have

∥u∥A =∑π∈G

dπ ∥u(π)∥S1

where ∥u(π)∥S1 is the trace-norm of the dπ × dπ-matrix u(π).An operator space structure on a given complex linear space X is an assignment

of norms, one on each space Mn(X) for natural number n, which satisfies certaincompatibility conditions; see, for example M1 and M2 of [3, p. 20]. We shall notrequire these explicit axioms here. Of importance to us, are the following facts. First,any von Neuman algebra V, in particular VN(G), will have assigned to each Mn(V)the unique norm which realises it as a von Neumann algebra. Maps between operatorspaces, Φ : X → Y are those maps whose matrix amplifications Φ(n) : Mn(X) →

188

Convolutions on the Haagerup tensor products of Fourier algebras

Mn(Y), Φ(n)[Xi j] = [Φ(Xi j)] are uniformly bounded: ∥Φ∥cb = supn

∥∥∥Φ(n)∥∥∥ < ∞.

The space CB(X,Y) of completely bounded maps is itself an operator space via theisometric identifications [Φi j] 7→ (X 7→ [Φi j(X)]) : Mn(CB(X,Y)) → CB(X,Mn(Y)),where Mm(Mn(X)) = Mmn(X). If V is a von Neumann algebra, then its predual V∗inherits the operator space structure from the inclusionV∗ →V∗. A map Φ : X → Yis called a complete isometry if each Φ(n) is an isometry, and a complete quotient ifeach Φ(n) is a quotient map. In the latter case we say Y is a complete quotient spaceof X. See [3, I.3] for details.

2.2. The Haagerup tensor product of Fourier algebras Fix a Hilbert spaceH . In[1, 10] it is shown that each weak*-weak* continuous completley bounded operator Φon B(H) — we shall write Φ ∈ CBσ(B(H)) — is given by

Φ(T ) =∑i∈I

ViTWi (3)

where Vi,Wii∈I is a family in B(H) for which each of the series∑

i∈I ViV∗i and∑i∈I W∗

i Wi is weak*-convergent. We shall write Φ =∑

i∈I Vi ⊗ Wi, accordingly.Furthermore, we have completely bounded norm

∥Φ∥cb = min

∥∥∥∥∥∥∥∑i∈I

ViV∗i

∥∥∥∥∥∥∥1/2 ∥∥∥∥∥∥∥∑i∈I

W∗i Wi

∥∥∥∥∥∥∥1/2

: Φ =∑i∈I

Vi ⊗Wi as in (3)

and operator composition∑

i∈I

Vi ⊗Wi ∑i′∈I

V ′i′ ⊗W ′i′ =

∑i∈I

∑i′∈I

ViV ′i′ ⊗W′i′Wi. (4)

Hence it is sensible to write

CBσ(B(H)) = B(H) ⊗w∗h B(H)

and call this space the weak* Haagerup tensor product.Let V ⊆ B(H) be a von Neumann algebra and V′ its commutant. It is shown in

[1, 10] that subspace of V′-bimodule maps in CBσ(B(H)) are exactly those elementwhich admit a representation as in (3) with each Vi,Wi in V. The description ofthe norm, with minimum taken over elements Vi and Wi form V, and the operatorcomposition are maintained, making this a closed subalgebra of B(H)⊗w∗h B(H). Wedenote this space by V ⊗w∗h V. Let V∗ denote the predual of V. We define for anelementary tensor u = v ⊗ w inV∗ ⊗ V∗ and Φ =

∑i∈I Vi ⊗Wi inV ⊗w∗h V, the dual

pairing⟨u,Φ⟩ =

∑i∈I

⟨v,Vi⟩⟨w,Wi⟩.

and define the Haagerup norm onV∗ ⊗V∗ by

∥u∥h = sup|⟨u,Φ⟩| : Φ ∈ V ⊗w∗h V, ∥Φ∥cb ≤ 1.We then let V∗ ⊗h V∗ denote the completion of V∗ ⊗ V∗ with respect to this norm.Then, as shown in [1] the dual pairing above extends to a duality

(V∗ ⊗h V∗)∗ V ⊗w∗h V. (5)

189


3. Third Section

Our main method for dealing with understanding Γ, as defined on either ofA(G)⊗A(G), or on A(G) ⊗h A(G), is to study its adjoint. To this end considerΓ : Trig(G) ⊗ Trig(G)→ Trig(G). If u, v ∈ Trig(G) and t ∈ G we have

⟨Γ(u ⊗ v), λ(t)⟩ =∫

Gu(s)v(t−1s) ds =

∫G⟨u ⊗ v, λ(s) ⊗ λ(t−1)λ(s)⟩ ds

and hence we have that Γ∗(λ(t)) =∫

G λ(s)⊗λ(t−1)λ(s) ds, in a weak* sense. By weak*-density of span λ(G) in Trig(G)′ we conclude that for T in Trig(G)′ we have

Γ∗(T ) =∫

Gλ(s) ⊗ Tλ(s) ds (6)

where the integral is understood in the weak* sense.We define A∆(G) = Γ(A(G)⊗A(G)). In [7] this was regarded as a quotient space of

A(G ×G) A(G)⊗A(G) and assigned a norm accordingly. We augment the concreterealisation of this norm, computed in [7, Thm. 2.2], by specifying the operator spacestructure on A∆(G) in a concrete manner.

Proposition 3.1. The Banach algebra A(G) ⊗h A(G) is semi-simple.

Theorem 3.2. The operator space structure on A∆(G), qua complete quotient ofA(G)⊗A(G) by Γ, is given by the weighted operator space direct sum

A∆(G) = ℓ1-⊕π∈G

d3/2π S2

dπ,r

where S2d,r denotes the d × d matrices with Hilbert-Schmidt norm and row operator

space structure.

The next result is a bit of a surprise. It shows that A(G) ⊗h A(G) behaves exactlyas does A(G)⊗A(G) with respect to Γ.

Theorem 3.3. We have that Γ(A(G) ⊗h A(G)) = A∆(G). Moreover, if A∆(G) is giventhe operator space structure in Theorem 3.2, above, then Γ : A(G) ⊗h A(G) → A∆(G)is a complete quotient map.

3.1. Convolution We now wish to consider the map Γ on A(G) ⊗h A(G). ConsiderΓ : Trig(G) ⊗ Trig(G)→ Trig(G). If u, v ∈ Trig(G) and t ∈ G we have

⟨Γ(u ⊗ v), λ(t)⟩ =∫

Gu(s)v(s−1t) ds =

∫G⟨u ⊗ v, λ(s) ⊗ λ(s−1)λ(t)⟩ ds

and hence we have that Γ∗(λ(t)) =∫

G λ(s)⊗λ(s−1)λ(t) ds, in a weak* sense. By weak*-density of span λ(G) in Trig(G)′ we conclude that for T in Trig(G)′ we have

Γ∗(T ) =∫

Gλ(s) ⊗ λ(s−1)T ds (7)

where the integral is understood in the weak* sense.

190

Convolutions on the Haagerup tensor products of Fourier algebras

Theorem 3.4. We have Γ(A(G)⊗hA(G)) = A(G). Moreover, Γ : A(G)⊗hA(G)→ A(G)is a complete quotient map.

Corollary 3.5. The convolution algebra (A(G), ∗) = (A(G), Γ) is completely isomor-phic to an operator algebra.

Let us close this section with a remark on convolution applied to A(G ×G).

Proposition 3.6. We have a completely isometric identification

Γ(A(G ×G)) = ℓ1-⊕π∈G

d2πS1

dπ

where we regard this space as a complete quotient space of A(G ×G) by Γ.

4. Spectral synthesis

Let us use Theorems 3.3 and 3.4 to observe some further connections betweenA(G × G) and A(G) ⊗h A(G), and also between V(G × G) and A(G) ⊗h A(G). Thisaddresses a question asked in [2, p. 21].

Proposition 4.1. Let θ, θ : G ×G → G be given by θ(s, t) = st−1 and θ(s, t) = st. Also,let E ⊂ G be closed. Then the following are equivalent:

(i) E is a set of spectral synthesis for A∆(G);(iii) θ−1(E) is a set of spectral synthesis for A(G) ⊗h A(G).

Also, the following are equivalent:(i’) E is a set of spectral synthesis for A(G);(iii’) θ−1(E) is a set of spectral synthesis for A(G) ⊗h A(G).

It is well-known, see for example [8], that point sets are spectral for A(G). Sinceθ−1(e) = (s, s−1) : s ∈ G, we gain the following.

Corollary 4.2. The anti-diagonal ∆ = (s, s−1) : s ∈ G is a set of spectral synthesisfor A(G) ⊗h A(G).

[1] D.P. Blecher and R.R. Smith. The dual of the Haagerup tensor product. J. London Math. Soc.(2), 45 (1992), no. 1, 126–144.

[2] M. Daws. Multipliers and abstract harmonic analysis. Talk at Canadian Abstract harmonicAnalysis Symposium, Saskatoon, 2010.See http://www1.maths.leeds.ac.uk/∼mdaws/talks/cahas hand.pdf.

[3] E.G. Effros and Z.-J. Ruan. Operator Spaces, volume 23 of London Math. Soc., New Series.Claredon Press, Oxford, 2000.

[4] E.G. Effros and Z.-J. Ruan. Operator space tensor products and Hopf convolution algebras. J.Operator Theory, 50 (2003), no. 1, 131–156.

[5] P. Eymard. L’algèbre de Fourier d’un groupe localement compact. Bull. Soc. Math. France, 92(1964), 181–236.

[6] B.E. Forrest. Fourier analysis on coset spaces. Rocky Mountain J. Math. 28 (1998), no. 1, 173–190.

191


[7] B.E. Forrest, E. Samei and N. Spronk. Convolutions on compact groups and Fourier algebras ofcoset spaces. Studia Math. 196 (2010), no. 3, 223–249.

[8] C. Herz. Harmonic synthesis for subgroups. Ann. Inst. Fourier (Grenoble) 23 (1973), no. 3,91–123.

[9] B.E. Johnson. Non-amenability of the Fourier algebra of a compact group. J. London Math. Soc.(2), 50 (1994), no. 2, 361–374.

[10] R.R. Smith. Completely bounded module maps and the Haagerup tensor product. J. Funct. Anal.,102 (1991), no. 1, 156–175.

[11] J. Tomiyama. Tensor products of commutative Banach algebras. Tôhoku Math. J. (2), 12 (1960),147–154.

M. Rostami,Department of Mathematics and Computer Sciences,Amirkabir University of Technology,Tehran, Irane-mail: [email protected]

N. Spronk,Department of Pure Mathematics„University of Waterloo,Ontario, Canadae-mail: [email protected]

192



January 20–21, 2016

Wavelets and amenability of a locally compact group

A. Bagheri Salec and J. Saadatmandan∗

Abstract

Let G be a locally compact group and π : G −→ U(Hπ) be a unitary representation. For each waveletη ∈ Hπ we have an isometry of Hπ into L2(G) .This isometry is interpreted of admissibility of thevector η in Hπ. The criterion of admissibility and the existence of admissible vector are vital subjectsin the wavelet theory. In this article we will study the existence of an admissible vector for irreduciblerepresentations and the amenability of G, In fact, it will be shown that if every irreducible representationof a locally compact group G has an admissible vector, then G is amenable.

2010 Mathematics subject classification: Primary 42C40, Secondary 43A65.Keywords and phrases: locally compact group, admissible vector, wavelet transform, amenability.

1. Introduction

The concept of wavelet are emerged more than 30 years ago and was exploded intoa wide range of applications such as digital storage of music and medical imaging.In this paper, regardless of applications, we study the relationship between waveletsand the amenability of the group G . By [2] we know that G is amenable if and only ifevery irreducible unitary representation of G is weakly contained in λG (the definitionis given in the next section). This motivates us to peruse on the relationship betweenamenability and existence of wavelets. In the next section, the definitions and basicrequirements are presented, and in the third section the main results are expressed.

2. Definitions and preliminaries

In this section, we present some definitions required in the article.

Definition 2.1. A unitary, strongly continuous representation, or simply a represen-tation, of a locally compact group G is a group homomorphism π : G −→ U(Hπ).which is continuous, when the right hand side is endowed with the strong operatortopology, that is x −→ π(x)η is continuous from G to Hπ for each η ∈ Hπ. Therefore,all coefficient functions of the type G ∋ x −→ ⟨ξ, π(x)η⟩ ∈ C are continuous, whereξ, η ∈ Hπ. This is because the weak and strong operator topologies coincide on U(Hπ).∗ speaker

193

A. Bagheri Salec and J. Saadatmandan

see [ [6] page 68 ] for detailes. The representation of G is called irreducible if Hπ and0 are the only subspace of that invariant under π . We use the notation π1 < π2 whenπ1 is unitary equivalent with some subrepresentation of π2 .

Definition 2.2. Let G be a locally compact group. The left regular representation λG

acts on L2(G) by (λG(x) f )(y) = f (x−1y). clearly this representation is unitary and forf , g ∈ L2(G), we have (g ∗ f ∗)(x) = ⟨g, λG(x) f ⟩, where f ∗(x) = f (x−1).

Definition 2.3. If π is a unitary representation of a locally compact group G on someHilbert space Hπ , it induces a *-representation π of L1(G) on Hπ via

⟨π( f )ξ, η⟩ =∫

Gf (x) ⟨ξ, π(x)η⟩ dx ( f ∈ L1(G), ξ, η ∈ Hπ)

which, in turn, extends to a *-representation π of the group C∗-algebra C∗(G) . see[[1], Chapter 2] for further details. In the case of G is amenable, for any boundedrepresentation of G on Hπ the extention π : C∗(G) −→ Hπ is bounded; see [8].If π1 and π2 are two unitary representations of G, we say that π1 is weakly containedin π2 if ker π2⊂ ker π1 In this case we use the notation π1 ≺ π2 .

Definition 2.4. Let (π,Hπ) denote a strongly continuous unitary representation of alocally compact group G. We endow G with it’s left Haar measure. For each η ∈ Hπ

associate the coeffitient operator Vη : Hπ −→ Cb(G) defined by Vηξ(x) = ⟨ξ, π(x)η⟩.The vector η ∈ Hπ is called admissible if Vη : Hπ −→ L2(G) is an isometery. In thiscase Vη : Hπ −→ L2(G) is called the (generalized) continuous wavelet transform.

Remark 2.5. We note that η is cyclic iff Vη, this time viewed as an operator Hπ −→Cb(G), is injective: Indeed, Vηϕ = 0 if ϕ ⊥ π(G)η, and that is equivalent to the factthat ϕ is orthogonal to the subspace spanned by π(G)η.

Definition 2.6. Let G be a locally compact group, and let E be a subspace of L∞(G)containing the constant functions. A mean on E is a functional m ∈ E∗ such that< 1,m >= ||m|| = 1. It is equivalent to < 1,m >= ||m|| = 1 and m is positive, i.e.< ϕ,m >≥ 0 (ϕ ∈ E , ϕ ≥ 0).

Definition 2.7. Let G be a locally compact group, and let E be a subspace of L∞(G)containing the constant functions and closed under complex conjugation. E is calledleft invariant if δg ∗ ϕ ∈ E for all ϕ ∈ E and g ∈ G. If E is left invariant, then a mean mon E is called left invariant if⟨

δg ∗ ϕ,m⟩= ⟨ϕ,m⟩ (ϕ ∈ E , g ∈ G).

Definition 2.8. A locally compact group G is amenable if there is a left invariant meanon L∞(G) .

194

Admidssiblity and Amenability

3. Main Results

In this section, we exhibit the main result of the article. We will not present acomplete proof here, but we give an address for detailes of proofs.

Lemma 3.1. If π1 and π2 are two unitary representations of G and π1 < π2 , then π1

is weakly contained in π2 .

Proof. By [2], Theorem 1.21 for π1 to be weakly contained in π2 it is sufficient thatfor every f ∈ L1(G) we have || f ||π1 ≤ || f ||π2 , where || f ||π = |||π( f )||| and |||π( f )||| denotesthe norm of the operator π( f ) in L(Hπ). if π1 < π2 then π1 is unitarily equivalent to asubrepresentation of π2 therefor to restriction of π2 to an invariant subspace M in Hπ2 ,scinse the norms of unitarily equivalent operators are equal then |||π1( f )||| ≤ |||π2( f )|||,therefor || f ||π1 ≤ || f ||π2 for every f ∈ L1(G), then π1 is weakly contained in π2.

Theorem 3.2. Let π be an irreducible representation of G. π has admissible vectorsiff π < λG .

Proof. If π has a cyclic vector η for which Vη is densely defined, there exists anisometric intertwining operator T : Hπ → L2(G). Hence, π < λG. Indeed, note thatby assumption Vη is densely defined, and it intertwines π and λG on its domain, seeProposition 2.16 and Theorem 2.25 in [7] for more detailes.

Theorem 3.3. Let G be a locally compact group. The following are equivalent:(i) G is amenable.(ii) Every irreducible, unitary representation of G is weakly contained in λG .(iii) The trivial representation of G on C (the one that maps every element of G to 1)is weakly contained in λG .

Proof. The equivalence of (i) and (ii) is proved by A. Hulanicki ([5]) whereas theequivalence of (ii) and (iii) is due to R. Godement ([3]).

Corollary 3.4. Let G be a locally compact group. If every irreducible representationof G has admissible vectors then G is amenable.

Proof. If π is an arbitrary irreducible representation which has admissible vectors,then by Theorem 3.2 π < λG . Therefore, by Lemma 3.1 π is weakly contained in λG

and by Theorem 3.3, G is amenable.

Remark 3.5. Let C∗λG(G) denote the reduced group C∗-algebra of G , i.e. the closure

of λG(L1(G)) with respect to the operator norm. Then conditions (i), (ii), and (iii)in Theorem 3.3are equivalent to C∗(G) C∗λG

(G) . Since C∗λG(G) is, in general,

much more understandable than C∗(G), C∗(G), though categorically ”nice” is hard to

195

A. Bagheri Salec and J. Saadatmandan

get hold in general, on the other hand C∗λG(G) is available a natural accessible C∗-

subalgebra of U(L2(G)). see [8] chapter 0 and chapter 4 for details. In addition λG

play the role for characterization of admissible vectors.

Remark 3.6. Note that amenability condition generally does not imply the existenceof admissible vector for all irreducible representations.

Example 3.7. Let G = R. Since G is an abelian group, it is amenable. But λG

has no irreducible subrepresentation, where λG is representation of R on L2(R)difined by [λG(x) f ](t) = f (t − x) . If there is an irreducible subrepresentation, thenits hilbert space would be one-dimensional, since G is abelian. Hence this space is ofthe form c f : c ∈ C for some, 0 , f ∈ L2(R) . Then for each x ∈ R we wouldhave f (t − x) = cx f (x) for some cx of modulus 1, so | f (t)| would be constant.This is impossible for f ∈ L2(R) unless f = 0. Hence there is some irreduciblerepresentation of R with no admissible vector.

Acknowledgement

We would like to express our gratitude to the committee members for taking timeout of their schedules for the articles arbitration and direction of petitioners. Specialthanks to organizers of the seminar.

[1] J. Dixmier, C∗-Algebras, North Holland, Amsterdam, 1977.[2] P. Eymard, L’algebre de Fourier d’un groupe localement compact, Bull. Soc. Math. France 92

(1964), 181-236.[3] R. Godement, Les fonctions de type positif et la tht’eorie des groupes , Trans. Amer. Math. Soc 63

(1948) 1-84.[4] A. Hulanicki, Means and Folner conditions on locally compact groups , Studia Math 24 (1964)87-

104.[5] A. Hulanicki, Groups whose regular representation weakly contains all unitary representations ,

Studia Math 24 (1964) 37-59.[6] G.B. Folland, A course in abstract harmonic analysis, CRC Press, Boca Raton, 1995.[7] H. Fuhr, Abstract Harmonic Analysis of Continuous Wavelet Transforms, Springer-Verlag Berlin

Heidelberg, 2005.[8] A. L. T. Paterson, Amenability, American Mathematical Society, 1998.[9] V. Runde, Lectures on Amenability, Lecture Notes in Mathematics 1774, Springer, 2002.

A. Bagheri Salec,Department of mathematics,University of Qom,City Qom, Irane-mail: [email protected]

J. Saadatmandan,Department of mathematics,

196

Admidssiblity and Amenability

University of Qom,City Qom, Irane-mail: [email protected]

197



January 20–21, 2016

Arens regularity of bounded bilinear maps and weaklycompactness

A. Sahleh and L. Najarpisheh∗

Abstract

Let X, Y and Z be Banach spaces. In this paper we study the relation between Arens regularity of abounded bilinear map m : X × Y −→ Z and weakly compactness of the linear operators representing m.

2010 Mathematics subject classification: Primary 46H25, Secondary 47A07.Keywords and phrases: Arens regularity, bounded bilinear map, weakly compact operator.

1. Introduction

For a normed space X we denote by X′ and X′′ the topological first and second dualsof X respectively. We consider X as naturally embedded into X′′. Let X, Y and Z benormed spaces and m : X × Y −→ Z a bounded bilinear map. In [1], R. Arens showedthat m has two natural but, in general, different extensions to the bilinear maps m′′′ andmr′′′ r. When m′′′ = mr′′′ r, the bilinear map m is said to be Arens regular. To establishour notation, we describe the construction briefly. The adjoint m′ : Z′ × X −→ Y ′ of mis defined by ⟨m′(z′, x), y⟩ = ⟨z′,m(x, y)⟩ for every x ∈ X, y ∈ Y, z′ ∈ Z′, which is also abounded bilinear map. By setting m′′ = (m′)′ and continuing in this way, the mappingm′′ : Y ′′ × Z′ −→ X′, m′′′ : X′′ × Y ′′ −→ Z′′ may be defined similarly.We also denote by mr the reverse map of m, that is the bounded bilinear mapmr : Y × X −→ Z defined by mr(y, x) = m(x, y) for every x ∈ X, y ∈ Y . mr maybe extended as above to mr′′′ r : X′′ × Y ′′ −→ Z′′.

Two natural extensions of the multiplication map π : A × A −→ A of a Banachalgebra (A, π), i.e. π′′′ and πr′′′ r, are actually the so-called first and second Arensproducts, which will be denoted by and , respectively. The Banach algebra (A, π)is said to be Arens regular if the multiplication map π is Arens regular. For exampleevery C∗ - algebra is Arens regular [2].

Let X and Y be normed spaces and T a bounded linear map from X to Y . DefineT ∗ : Y ′ −→ X′ by T ∗(y′) = y′oT , then T ∗ is a bounded linear map. T ∗ is called theadjoint or transpose of T .

∗ speaker

198

A. Sahleh and L. Najarpisheh

Now let X, Y and Z be Banach spaces. In [4] the bounded bilinear formsm : X ×Y −→ C were represented as m(x, y) = ⟨u(x), y⟩ = ⟨ν(y), x⟩ where u : X −→ Y ′

and ν : Y −→ X′ are continuous linear operators with ∥ u ∥=∥ ν ∥=∥ m ∥. A continuouslinear operator u : X −→ Y is said to be weakly compact if the image of the unit ballof X under u is relatively compact with respect to the weak topology of Y . Now letm : X × Y −→ C be a bilinear form and u (resp. ν) be the linear operator representingm. It is easy to show that u is weakly compact if and only if ν is. We also have m isArens regular if and only if u (resp. ν) is weakly compact [4, theorem 2.2].

Throughout this paper, X, Y and Z are arbitrary Banach spaces. We study the abovetheory for the bounded bilinear forms from X × Y to a Banach space Z instead of thefield of complex numbers.

2. Arens regularity and weakly compactness

In this section B(X,Y) is the space of bounded linear operators from X into Y .Let m : X × Y −→ Z be a bounded bilinear map. So it can be represented asm(x, y) = ⟨u(x), y⟩ = ⟨ν(y), x⟩ where u : X −→ B(Y,Z) and ν : Y −→ B(X,Z)are bounded linear operators. In Theorem 2.2, we prove that weakly compactnessof u (resp. ν) implies Arens regularity of m. Moreover, in Example 2.3, we showthat weakly compactness of u doesn’t imply the weakly compactness of ν and vice-versa. Finally, we show that in general, the converse of Theorem 2.2 is not correct (seeExample 2.5).

Proposition 2.1. [3, theorem 2.1] For a bounded bilinear map f : X × Y −→ Z thefollowing statements are equivalent:

(i) f is Arens regular;(ii) f ′′′′ = f r′′′′′ r;(iii) f ′′′′(Z′, X′′) ⊆ Y ′;(iv) the linear map x 7−→ f ′(z′, x) : X −→ Y ′ is weakly compact for every z′ ∈ Z′.

Theorem 2.2. Let the bounded linear operator u (resp. ν) representing m be weaklycompact. Then m is Arens regular.

Example 2.3. Let H be a infinite dimensional Hilbert space and m : B(H) × H −→ Hthe bounded bilinear map defined by m(T, h) = T (h) for all T ∈ B(H) and h ∈ H. Sou : B(H) −→ B(H), the linear operator representing m, is the identity operator andidentity can not be weakly compact unless B(H) be finite dimensional. Moreover forν : H −→ B(B(H),H), since H is reflexive the linear operator ν∗ : B(B(H),H)′ −→ His weakly compact. Thus weakly compactness of u is not equal to weakly compactnessof ν.

Remark 2.4. Let f : X × Y −→ Z be a bounded bilinear map. If Y is reflexive, then Y ′

is reflexive. So f ′′′′(Z′, X′′) ⊆ Y ′ and Proposition 2.1 implies that f is Arens regular.

199

Arens regularity and weakly compactness

Example 2.5. For H and m introduced in example 2.3 u is not weakly compact, whileby applying the above remark reflexivity of H implies the regularity of m. So Arensregularity of the bounded bilinear map m doesn’t imply weakly compactness of thebounded linear operator u representing m.

[1] R. Arens, The adjoint of a bilinear operation, Proc. Amer. Math. Soc. 2 (1951), 839-848.[2] P. Civin and B. Yood, The second conjugate space of a Banach algebra as an algebra, Pacific J.

Math. 11 (1961), 847-870.[3] S. Mohammadzadeh and H.R.E. Vishki, Arens regularity of module actions and the second adjoint

of a derivation, Bull. Austral. Math. Soc. 77 (2008), 465-476.[4] A. Ulger, Weakly compact bilinear forms and Arens regularity , J. Funct. Anal. 37 (1980), 697-

7o4.

A. Sahleh,Faculty of Mathematical Sciences, University of Guilan,Rasht, Irane-mail: [email protected]

L. Najarpisheh,Faculty of Mathematical Sciences, University of Guilan,Rasht, Irane-mail: [email protected]

200



January 20–21, 2016

An Application of Fourier Transform in the Study of Projectionsof Probability Measures

E. Salavati

Abstract

Characterizations of distributions using their lower-dimensional projections has been the subject of manyresearch works. One of the properties of the stable distributions is the existence of a multivariatedistribution which its one-dimensional projections are also stable. In this article we study the sameproperty for other distributions. We formulate the concept of n-dimensional lift and then prove somenecessary and sufficient conditions for its existence. The main tool in our study is the Fourier transformof measures.

2010 Mathematics subject classification: Primary 60Exx, Secondary 42A38.Keywords and phrases: Characteristic Function, Projection of Measures.

1. Introduction

Among distributions, the class of stable distributions have the interesting property thatthere exists a multivariate distribution, whose all of its linear combinations are Stable.This property makes the stable distributions computationally efficient.

The generalizing this projection property has been done in several ways. Forexample, [1] has derived conditions under which, a distribution whose all of itsprojections are stable, is stable.

One could ask if for a given non-stable distribution such multivariate distributionsexist at all. In this article we study this problem for general distributions.

We begin with the uniform distribution in section 2 we define the n-dimensionallift of a distribution. In Theorem 2.1 and 2.2 we prove that for uniform distribution,this concept exists only in 2 and 3 dimensions.

In Section 3 we consider the same problem for general distributions and providenecessary and sufficient conditions for it.

We end the Introduction by the definition of n-dimensional lift.

Definition 1.1. Let ρ be a given distribution on R. By an n-dimensional lift of ρ, wemean a distribution on Rn whose one-dimensional projections equal to ρ.

201

E. Salavati

2. Uniform distribution

In this section we study the exitence of n-dimensional lifts for uniform distribution.We will show that in dimensions 2 and 3 it exists but for n ≥ 4 it does not exist. We callan n-dimensional lift of the uniform distribution, a projectionally uniform distribution.

2.1. Dimensions 2 and 3

Theorem 2.1. For n = 2, 3, the projectionally uniform distribution on Rn exists.

Proof. It suffices to prove the statement for n = 3, since then the projection of thedistribution on x − y plane would satisfy the condition for n = 2.

For n = 3, let µ be the uniform distribution on the surface of a 3-sphere. We claimthat the projection of µ on any direction is a uniform distribution.

Let x = (x1, x2, x3) be the standard coordinate on R3 and let S 2 be the surface ofthe unit sphere:

S 2 = x : x21 + x2

2 + x23 = 1

Since µ is rotationally invariant, it suffices to prove the claim for just one direction,say, x1 direction.

Note that µ, the surface measure of S 2 can be written in coordinates as

dµ =1

4πx1dx2dx3

Let π : R3 → R be the projection π(x) = x1. We show that π∗µ is a uniformdistribution on [−1, 1]. For this, we compute

π∗µ([1 − a, 1]) =∫

x1∈[1−a,1]dµ =

14π

∫x1∈[1−a,1]

1x1

dx2dx3

Now, write the last integral in the polar coordinates for (x2, x3),

=1

4π

∫x1∈[1−a,1]

1x1

rdrdθ =12

∫x1∈[1−a,1]

rdrx1

By change of variable r =√

1 − x21,

=12

∫ 1

1−adx1 =

a2

Which shows that π∗µ is uniform on [−1, 1].

202

On Projections of Probability Measures

2.2. Dimension n ≥ 4 In order to prove the non-existence of projectionally uniformdistribution for n ≥ 4, we study some elementary properties of these distributions.

Let µ be a projectionally uniform distribution in Rn.Since the projection of µ on any direction is non-degenerate, hence its covariance

matrix is non-degenerate. So, by applying a linear transformation we may assume thatthe covariance matrix of µ is 1

3 I, i.e∫xxTµ(dx) =

13

I

This implies that for any u ∈ Rn,∫∥u.x∥2µ(dx) =

13∥u∥2 (1)

Now let X be a random vector with distribution mu. By assumption, u.X is auniform distribution with mean zero and by equation (1), it’s variance is 1

3∥u∥2, henceit should have a uniform distribution on [−∥u∥, ∥u∥].

Now, we can compute the characteristic function (Fourier transform) of µ,

ϕµ(u) =∫

eiu.xµ(dx) = E(eiu.X)

Now, note that the characteristic function of uniform distribution on [−a, a] is sin(ta)ta ,

hence

ϕµ(u) =sin ∥u∥∥u∥

We are ready to prove the theorem.

Theorem 2.2. For n ≥ 4, there is no projectionally uniform distribution on Rn.

Remark 2.3. As an application of the projectionally uniform distribution introducedin this section, we show how it can be used to generate correlated uniform variableswith prescribed covariance matrix.

Let X be a uniform point on the surface of S 2. If X = (X1, X2, X3), then bysymmetry, X1, X2 and X3 have mean zero and covariance matrix 1

3 I. Hence, for any3 × 3 matrix A, AX will be a random vector with covariance matrix 1

3 AAT .Therefore if we are given a covariance matrix C, we can put A = (3C)

12 such that

13 AAT = C and then AX gives us three uniform variables with covariance matrix C.

3. General Distributions

In this section, we consider a general measure ρ on R and study the existence of itsn-dimensional lift.

We begin with a necessary and sufficient condition in dimension n = 3.

203

E. Salavati

Theorem 3.1. If ρ is symmetric around 0 and has compact support, then the necessaryand sufficient condition for existence of a 3-dimensional lift is that the distributionfunction F(t) = ρ((t,∞)) be decreasing and convex for t ∈ [0,∞).

Now, We state and prove a necessary condition for compact support measures onR, to have an n-dimensional lift.

Theorem 3.2. Let ρ be a symmetric and compact support measure on R with essentialsupremum a. If ρ has an n-dimensional lift then for each ϵ > 0,

ρ((a − ϵ, a]) ≤ c(ϵ

a)

n−12

where c is a constant which depends only on n.

Proof. We follow the steps of the proof from [3].ρ is supported in [−a, a]. For simplicity, assume that a = 1. Assume that µ is an

n-dimensional lift for ρ.Let X be a random vector chosen with distribution µ and let V be a uniformly

chosen random unit vector in Rn, independent of X.We compute the probability p = P(V.X > 1 − ϵ) in two ways.First, fix V , and note that V.X has the distribution ρ, hence p = ρ((1 − ϵ,∞)).Now, fix X and note that P(V.X > 1− ϵ) is increasing in ∥X∥, hence it is maximized

when X is a unit vector, in which case, P(V.X > 1 − ϵ) is the proportion of a sphericalcap on S n−1 with depth ϵ, which is less than cϵ

n−12 , for a suitable c. Hence, p ≤ cϵ

n−12 .

Remark 3.3. Theorem 3.2 provides another proof for the fact that the uniform distri-bution does not have an n-dimensional lift for n ≥ 4.

Remark 3.4. Finally, we note that in cases that n-dimensional lift exists, one cancompute them using Hankel transform, which is the Fourier transform for rotationallysymmetric distributions. More precisely, let ρ be a given distribution on R withcharacteristic function (Fourier transform) ϕ(u), then with the same arguments as insection 1, we will have

ϕµ(u) = ϕ(∥u∥)

and the main problem is whether ϕ(∥u∥) is the characteristic function of a distributionor not. In case that it is the characteristic function of a continuous distribution, it’sdensity can be computed using the Hankel transform,

f (x) = x1− n2 (2π)

n2

∫ ∞

0r

n2ϕ(r)J n

2−1(xr)dr

where Jν is the Bessel function of the first kind of order ν.

204

On Projections of Probability Measures

[1] Gupta, Arjun K., Truc T. Nguyen, and Wei-Bin Zeng. "Conditions for stability of laws with allprojections stable." Sankhy: The Indian Journal of Statistics, Series A (1994): 438-443.

[2] Rudin, W., Fourier analysis on groups, Wiley-Interscience, 1990.[3] MathOverflow, A probability distribution in n dimensional space which its projection on any line

is a uniform distribution?, URL:http://mathoverflow.net/q/210041.

E. Salavati,School of Mathematics,Institute for Research in Fundamental Sciences (IPM),Tehran, Irane-mail: [email protected]

205



January 20–21, 2016

Fusion frame multipliers

M. Shamsabadi∗ and A. A. Arefijamaal

Abstract

Frame multipliers are interesting mathematical objects and they are also important tools for applications.In this paper, we will focus on fusion frame multipliers and investigate representation of their inverses.

2010 Mathematics subject classification: Primary 41A58, Secondary 42C15.Keywords and phrases: fusion frame; multiplier; inverse multiplier; canonical dual; Riesz basis.

1. Introduction

Nowadays, applications of signal processing play an important role in numeroustechnical items. In these applications time-invariant filters i.e. convolution operatorsare used very often. Frame multipliers are applied for example in psychoacousticalmodeling [6] and denosing [12]. Multipliers are a particular way to implement time-variant filters [13]. In this respect, it is important to find the inverse of a multiplierif it exists. Moreover, a question is how to express the inverse of an invertible framemultiplier as a multiplier [7].

Throughout this paper H denotes a separable Hilbert space and I a countableindex set. The identity operator on H is denoted by IH . A sequence m := mii∈I

of complex scalars is called semi-normalized if there exist constants a, b such that0 < a ≤ |mn| ≤ b < ∞, for all n ∈ N.

For two sequencesΦ := ϕii∈I andΨ := ψii∈I in a Hilbert spaceH and a sequencem of complex scalars, the operatorMm,Φ,Ψ : H → H given by

Mm,Φ,Ψ f =∑

mi⟨ f , ψi⟩φi, ( f ∈ H),

is called a multiplier. The sequence m is called symbol. If Φ and Ψ are Bessel se-quences forH and m ∈ ℓ∞, thenMm,Φ,Ψ is well defined and ∥Mm,Φ,Ψ∥ ≤

√BΦBΨ∥m∥∞,

see [2, 4]. The invertibility of multipliers, which plays a key role in the topic, is dis-cussed in [4, 7, 15].

∗ speaker

206

M. Shamsabadi and A. A. Arefijamaal

2. Fusion frame

In this section, we review some definitions and primary results of fusion frames.For more informations see [8, 9]. Throughout this paper, πW denotes the orthogonalprojection fromH onto a closed subspace W.

Definition 2.1. Let Wii∈I be a family of closed subspaces ofH and ωii∈I be a familyof weights, i.e. ωi > 0, i ∈ I. The sequence (Wi, ωi)i∈I is called a fusion frame forHif there exist constants 0 < A ≤ B < ∞ such that

A∥ f ∥2 ≤∑i∈I

ω2i ∥πWi f ∥2 ≤ B∥ f ∥2, ( f ∈ H).

The constants A, B are called the fusion frame bounds. If we only have the upperbound, we call (Wi, ωi)i∈I a Bessel fusion sequence. A fusion frame is called tight, ifA, B can be chosen to be equal, and Parseval if A = B = 1. If ωi = ω for all i ∈ I,the collection (Wi, ωi)i∈I is called ω-uniform and we abbreviate 1- uniform fusionframes as Wii∈I . A fusion frame (Wi, ωi)i∈I is said to be an orthonormal fusion basisif H =

⊕i∈I Wi and it is called Riesz decomposition of H if for every f ∈ H , there

is a unique choice of fi ∈ Wi such that f =∑

i∈I fi. It is clear that every orthonormalfusion basis is a Riesz decomposition for H , and also every Riesz decomposition is a1-uniform fusion frame forH .

Every family Wii∈I of closed subspaces of H is an orthonormal basis if and onlyif is a 1-uniform parseval fusion frame , see[8].

Recall that for each sequence (Wi, ωi)i∈I of closed subspaces inH , the space

(∑i∈I

⊕Wi)ℓ2 = fii∈I : fi ∈ Wi,

∑i∈I

∥ fi∥2 < ∞,

with the inner product

⟨ fii∈I , gii∈I⟩ =∑i∈I

⟨ fi, gi⟩,

is a Hilbert space. The synthesis operator TW : (∑

i∈I⊕

Wi)ℓ2 → H for a Besselfusion sequence (Wi, ωi)i∈I is defined by

TW( fii∈I) =∑i∈I

ωi fi, ( fii∈I ∈∑i∈I

⊕Wi).

Its adjoint operator T ∗W : H → (∑

i∈I⊕

Wi)ℓ2 which is called the analysis operator isgiven by

T ∗W( f ) = ωiπWi ( f ), ( f ∈ H).

If (Wi, ωi)i∈I is a fusion frame, the fusion frame operator S W : H → H , which isdefined by S W( f ) = TWT ∗W( f ) =

∑i∈I ω

2i πWi ( f ), is bounded, invertible and positive.

Hence, we have the following reconstruction formula [8]

f =∑i∈I

ω2i S −1

W πWi f , ( f ∈ H),

207


The family (S −1W Wi, ωi)i∈I , which is also a fusion frame, is called the canonical dual

of (Wi, ωi)i∈I and satisfies the following reconstruction formula

f =∑i∈I

ω2i πS −1

W WiS −1

W πWi f , ( f ∈ H).

In general, every Bessel fusion sequence (Vi, υi)i∈I is called a dual of (Wi, ωi)i∈I , if

f =∑i∈I

ωiυiπVi S−1W πWi f , ( f ∈ H),

see [8, 9] for more details. In [10], it is shown that a Bessel fusion sequence (Vi, υi)i∈Iis a dual of fusion frame (Wi, ωi)i∈I if and only if TVϕvwT ∗W = IH , where the boundedoperator ϕvw : (

∑i∈I

⊕Wi)ℓ2 → (

∑i∈I

⊕Vi)ℓ2 is given by

ϕvw( fii∈I) = πVi S−1W fii∈I

If Wii∈I is a family of closed subspaces of H and ωii∈I is a family of weightsthen we say that (Wi, ωi)i∈I is a fusion Riesz basis forH if spani∈IWi = H and thereexist constants 0 < C ≤ D < ∞ such that for each finite subset J ⊆ I

C

∑j∈J

∥ f j∥2

1/2

≤ ∥∑j∈J

ω j f j∥ ≤ D

∑j∈J

∥ f j∥2

1/2

, ( f j ∈ W j).

Theorem 2.2. [8] Let Wii∈I be a fusion frame for H and ei j j∈Ji be an ONB for Wifor each i ∈ I. Then the following conditions are equivalent:(1) Wii∈I is a Riesz decomposition ofH .(2) The synthesis operator TW is one to one.(3) The synthesis operator T ∗W is onto.(4) Wii∈I is a fusion Riesz basis forH .(5) ei ji∈I, j∈Ji is a Riesz basis forH .

Theorem 2.3. Let Wii∈I be a fusion frame inH . Then the following are equivalent:(1) Wii∈I is a fusion Riesz basis.(2) S −1

W Wi ⊥ W j for all i, j ∈ I, i , j.(3) πWi S

−1W πW j = δi jπW j for all i, j ∈ I.

3. Fusion frame multiplierThe frame multipliers were first introduced by Balazs, [4] and developed by many

authors [2, 4, 7]. In this section, we establish fusion frame multipliers and investigatetheir properties.

Definition 3.1. Let (Wi, ωi)i∈I and (Vi, υi)i∈I be fusion frame and Bessel fusionsequence, respectively. Define Mm,W,V : H → H by

Mm,W,V f =∑i∈I

miωiυiπVi S−1W πWi f ,

where m = mi∞i=1 ∈ ℓ∞.

208

M. Shamsabadi and A. A. Arefijamaal

Lemma 3.2. Let (Wi, ωi)i∈I be a fusion frame. If (Vi, υi)i∈I is a Bessel fusionsequence and m ∈ ℓ∞, then(1) M∗m,W,V = Mm,V,W .(2) ∥Mm,W,V∥ ≤ supi∈I |mi|∥S −1

W ∥√

BW BV .

Now, we state the fusion frame multipliers with respect to frame multipliers.

Lemma 3.3. Let (Wi, ωi)i∈I and (Vi, υi)i∈I be fusion frame and Bessel fusion se-quence, respectively. Then

Mm,W,V =Mm,G,F , (1)

in particular,M1,W,V = TGT ∗F = TVϕvwT ∗W , (2)

where F := ωiπWi S−1W e ji, j, G := υiπVi e ji, j are the local frames of (Wi, ωi)i∈I and

(Vi, υi)i∈I , respectively.

Proposition 3.4. M1,W,W = S W if Wii∈I is a Riesz fusion basis.

Corollary 3.5. Wii∈I is a Riesz fusion basis if and only if it’s not tight non-parseval.In particular, Wii∈I is Riesz fusion basis and dual of itself if and only if Wii∈I is anorthonormal basis.

Note that if M1,W,W is invertible, it’s not necessary Wii∈I is Riesz fusion basis. Forexample, let Wii∈I be tight non-parseval. By last corollary Wii∈I is not Riesz fusionbasis but

M1,W,W = S −1W

∑i∈I

πWi = IH .

Theorem 3.6. Let Wii∈I and Zii∈I be two fusion frames and Vii∈I and Hii∈I be twoBessel fusion sequences such that Wii∈I and Hii∈I are biorthogonal. Then

M1,W,VM1,Z,H =M1,πVi S−1W πWiπHi e ji, j,πZi S

−1Z e ji, j .

Corollary 3.7. Let Wii∈I be an orthogonal Riesz fusion basis and Vii∈I be a fusionframe. Then(1) M1,W,WM1,V,W = M1,V,W .(2) If M1,V,W is invertible, then Wii∈I is an orthonormal fusion basis.

[1] A. A. Arefijamaal, F. Arabyani, Constructing dual and approximate dual fusion frame, Submitted.[2] M.L. Arias, M. Pacheco, Bessel fusion multipliers, J. Math. Anal. Appl. 348(2008)581-588.[3] A. Rahimi, P. Balazs, Multipliers for p- Bessel sequence in Banagh spaces, Integral equations

Operator Theory 68 (2010) 193-205.[4] P. Balazs, Basic definition and properties of Bessel multipliers, J. Math. Anal. Appl. 325 (1)(2007)

571-585.[5] P. Balazs, D. Bayer, A. Rahimi, Multipliers for continuous frames in Hilbert spaces, J. Phys. A 45

(24) (2012) 244023, 20 pp.

209


[6] P. Balazs, B. Laback , G. Eckel, W.A. Deutsch, Time-frequency sparsity by removing perceptuallyirrelevant components using a simple model of simultaneous masking, IEEE Trans. Audio SpeechLang. Processing 18 (1) (2010) 34-49

[7] P. Balazs, D.T. Stoeva, Representation of the inverse of a frame multiplier, J. Math. Anal. Appl.422 (2015) 981-994.

[8] P.G. Cassaza, G. Kutyniok, Frames of subspaces, Contemp. Math. vol 345, Amer. Math. Soc.Providence, RI, 2004, pp. 87-113

[9] P. Gavruta, On the duality of fusion frames, J. Math. Anal. Appl. vol 333 (2007) 871-879.[10] E. Osgooei, A.A. Arefijamaal, Compare and contrast between duals of fusion and discrete frames,

Submitted.[11] O. Christensen, An introduction to frames and Riesz Bases, Birkhäuser, Boston, 2003.[12] P. Majdak, P. Balazs, W. Kreuzer, M. Dorfler, A time-frequency metod for increasing the signal-

to-noise ratio in system identification with exponential sweeps, in:Proceedings of the 36th IEEEinternational Conference on Acoustics, Speech, and Signal Processing, ICASSP 2011, 2011, pp.3812-3815

[13] G. Matz, F. Hlawatsch, Linear time-frequency filters: online algoritms and applications, in: A.Papandreou-Suppappola (Ed.), Applications in Time-Frequency Signal Processing, CRC Press,Boca Raton, Fl, 2002, pp. 205-271 (Ch. 6).

[14] D. T. Stoeva, P. Balazs, Weighted frames and frame multipliers, in: Annual of the university ofArchitecture, Civil Engineering and Geodsey , vols. XLIII-XLIV 2004-2009, 2012, 33-42.

[15] D.T. Stoeva, P. Balazs, Invertiblity of multipliers, Appl. comput. Harmon. Anal. 33 (2) (2012)292-299.

[16] D.T. Stoeva, P. Balazs, Canonical forms of unconditionally convergent multipliers, J. Math.Anal.Appl. 399 (1) (2013) 252-259.

[17] D.T. Stoeva, P. Balazs, Detailed characterization of unconditional convergence and invertibility ofmultipliers, Sample. Theory Signal Image Process. 12 (2) (2013) 87-125.

[18] D.T. Stoeva, P. Balazs, Riesz Bases multipliers, Concrete Operators, Spectral Theory, Operators inHarmonic Analysis and Approxmation Operator Theory: Advances and Applications, 236(2014)475-482.

M. Shamsabadi,Department of mathematics,University of Hakim Sabzevari,City Sabzevar, Irane-mail: [email protected]

A. A. Arefijamaal,Department of mathematics,University of Hakim Sabzevari,City Sabzevar, Irane-mail: [email protected]

210



January 20–21, 2016

Some properties of weighted composition operators on L2(Σ)

S. Shamsi Gamchi

Abstract

In this note unbounded hyperexpansive weighted composition operators are investigated. As a conse-quence unbounded hyperexpansive multiplication and composition operators are characterized. Someexamples are provided to illustrate concrete application of the main results of the paper.

2010 Mathematics subject classification: 47B38.Keywords and phrases: weighted composition operator, hyperexpansive operators, unbounded operator.

1. Introduction

Weighted composition operators are a general class of operators and they appear nat-urally in the study of surjective isometries on most of the function spaces, semigrouptheory, dynamical systems, Brennan’s conjecture, etc. This type of operators are ageneralization of multiplication operators and composition operators. The basic prop-erties of unbounded weighted composition operators on L2(Σ) are studied by PiotrBudzynski, Zenon Jan Jablonski, Il Bong Jung and Jan Stochel [1] and some othermathematicians. In this paper we consider unbounded weighted composition opera-tors on the Hilbert space L2(Σ) and study hyperexpansive weighted composition op-erators. As a consequence hyperexpansive multiplication and composition operatorsare characterized. The results of this paper extends some results of [3]. Let H bestand for a Hilbert space. By an operator on H we understand a linear mappingT : D(T ) ⊆ H → H defined on a linear subspace D(T ) of H which is called thedomain of T . Set D∞(T ) = ∩∞n=1D(T n). Given an operator T on H , we define thegraph norm ∥.∥T onD(T ) by

∥ f ∥2T = ∥ f ∥2 + ∥T f ∥2, f ∈ D(T ).

The next proposition can be easily deduced from the closed graph theorem.

Proposition 1.1. If T is a closed operator onH such that T (D(T ) ⊆ (D(T ), then T isa bounded operator on the Hilbert space (D(T ), ∥.∥T ).

211

S. Shamsi Gamchi

2. Second section

For a sub-σ-finite algebra A ⊆ Σ, the conditional expectation operator associatedwith A is the mapping f → EA f , defined for all non-negative f as well as for allf ∈ L2(Σ), where EA f , by the Radon-Nikodym theorem, is the unique A-measurablefunction satisfying

∫A f dµ =

∫A EA f dµ for all A ∈ A. As an operator on L2(Σ),

EA is an idempotent and EA(L2(Σ)) = L2(A). If there is no possibility of confusionwe write E( f ) in place of EA( f ) [5]. For a complex Σ-measurable function u onX. Define the measure µu : Σ → [0,∞] by µu(E) =

∫E |u|

2dµ, E ∈ Σ. By theRadon-Nikodym theorem, if µu ϕ−1 ≪ µ, then there exists a unique (up to a.e. µequivalence) Σ-measurable function J : X → [0,∞] such that µu(ϕ−1(E)) =

∫E Jdµ. If

µ ϕ−1 ≪ µ, then µu ϕ−1 ≪ µ. So, we get that J = hE(|u|2) ϕ−1, where h = dµϕ−1

dµ .If ϕ is a non-singular transformation, then the operator uCϕ is well-defined. Well-defined operators of the form uCϕ( f ) = u. f ϕ acting in L2(µ) = L2(X,Σ, µ) withD(uCϕ) = f ∈ L2(µ) : u. f ϕ ∈ L2(µ) are called weighted composition operators. Ifµ ϕ−1 ≪ µ, then for every f ∈ D(uCϕ) we have |uCϕ( f )∥2 =

∫X hE(|u|2) ϕ−1| f |2dµ.

By induction we get that for every n ≥ 1 ∥(uCϕ)n( f )∥2 =∫

X Jn| f |2dµ, for allf ∈ D((uCϕ)n). Where uϕ,n = u.u ϕ.u ϕ2...u ϕn−1, Jn = hE(Jn−1|u|2) ϕ−1, his the Radon-Nykodim derivative dµϕ−1

µ, E is conditional expectation with respect to

ϕ−1(Σ) and J0 = 1.

Lemma 2.1. Let w = 1 + J and dν = wdµ. Then we have• S (w) = X and L2(ν) = D(uCϕ),

And also, the followings are equivalent;

• uCϕ is densely defined.

• J < ∞ a.e. µ.

If all functions Ji = hE(Ji−1|u|2) ϕ−1, i = 1, ..., n, are finite valued, where hi is theRadon-Nykodim derivative dµϕ−i

dµ , then we set

J,n(x) =∑

0≤i≤n

(−1)i(

ni

)Ji(x).

Proposition 2.2. If D(uCϕ) is dense in L2(Σ), then the following conditions areequivalent:• uCϕ(D(uCϕ)) ⊆ D(uCϕ).

• There exists c > 0 such that J2 ≤ c(1 + J1) a.e. µ.

212

WEIGHTED COMPOSITION OPERATORS

3. Hyperexpansive weighted composition operators

In this section we are going to present conditions under which EMu is k-isometry,k-expansive, k-hyperexpansive and completely hyperexpansive. First we recall someconcepts that we need them in the sequel. For an operator T on Hilbert space H weset

ΘT,n( f ) =∑

0≤i≤n

(−1)i(

ni

)∥T i( f )∥2, f ∈ D(T n), n ≥ 1.

We recall that an operator T onH is:• k-isometry (k ≥ 1) if ΘT,k( f ) = 0 for f ∈ D(T k),

• k-expansive (k ≥ 1) if ΘT,k( f ) ≤ 0 for f ∈ D(T k),

• k-hyperexpansive (k ≥ 1) if ΘT,n( f ) ≤ 0 for f ∈ D(T n) and n = 1, 2, ..., k.

• completely hyperexpansive if ΘT,n( f ) ≤ 0 for f ∈ D(T n) and n ≥ 1.For more details one can see [2–4]. Now we present our main results as follows.

Proposition 3.1. IfD((uCϕ)n) is dense in L2(µ) for a fixed n ≥ 1, then:• uCϕ is k-expansive if and only if J,n(x) ≤ 0 a.e. µ.

• uCϕ is k-isometry if and only J,n(x) = 0 a.e. µ.

Proposition 3.2. IfD((uCϕ)2) is dense in L2(µ) and uCϕ is 2-expansive, then:• uCϕ leaves its domain invariant:

• Jk ≥ Jk−1 a.e. µ for all k ≥ 1.

Recall that a real-valued map φ on N is said to be completely alternating if∑0≤i≤n(−1)i

(ni

)φ(m + i) ≤ 0 for all m ≥ 0 and n ≥ 1. The next theorem is a

direct consequence of proposition 2.3 and 2.4.

Theorem 3.3. IfD((uCϕ)2) is dense in L2(µ) and k ≥ 1 is fixed, then:• uCϕ is k-hyperexpansive if and only if J,i(x) ≤ 0 a.e. µ for i = 1, ..., k.

• uCϕ is completely hyperexpansive if and only if Ji∞i=0 is a completely alternatingsequence for almost every x ∈ X.

Theorem 3.4. Let uCϕ be 2-expansive operator and• Let (X,Σ, µ) is an infinite measure space such that µu(X) < ∞ and D((uCϕ)2) is

dense in L2(µ).

213

S. Shamsi Gamchi

• Let (X,Σ, µ) is a measure space such that µu(X) < ∞, u ≤ 1 a.e. µ andD((uCϕ)2)is dense in L2(µ).

If one of above two conditions holds, then uCϕ is an isometry. Also, if uCϕ is denselydefined, u , 0 a.e. µ and the sigma algebra ϕ−1(Σ) is essentially all of Σ, with respectto µ, then uCϕ is a unitary operator.

[1] Piotr Budzynski, Zenon Jan Jablonski, Il Bong Jung, Jan Stochel, Unbounded Weighted Com-position Operators in L2-Spaces, arXiv:1310.3542.

[2] Z. J. Jablonski. Complete hyperexpansivity, subnormality and inverted boundedness conditions,Integral Equations Operator Theory 44 (2002) 316-336.

[3] J. Jablonski, Hyperexpansive composition operators, Math. Proc. Camb. Phil. Soc. 135 (2003),513-526.

[4] J. Jablonski and J. Stochel, Unbounded 2-hyperexpansive operators, Proc. Edinburgh Math. Soc.44 (2001), 613-629.

[5] M. M. Rao, Conditional measure and applications, Marcel Dekker, New York, 1993.

S. Shamsi Gamchi,Department of mathematics,Payame noor university,Tehran, Irane-mail: [email protected]

214



January 20–21, 2016

On the cohomological properties of dual Banach algebras

A. Shirinkalam∗ and A. Pourabbas

Abstract

Some cohomological properties of dual Banach algebras are studied. We introduce the notion ofConnes-biprojective dual Banach algebras. We investigate the relation between this notion and Connes-amenability. In paticular, for a given dual Banach algebraA, we show thatA is Connes-amenable if andonly if it is Connes-biprojective and has a bounded approximate identity.


Connes-amenability was first considered for von Neumann algebras in [8] in the sensethat it takes the dual space structure of a von Neumann algebra into account. Thereason why this notion of amenability is associated with A. Connes are his papers [2]and [3]. Connes-amenability is equivalent to several other important properties of vonNeumann algebras, such as injectivity and semidiscreteness([13], [2], [3], [1]).

In [10], Runde extended the notion of Connes-amenability to the larger class ofBanach algebras, the dual Banach algebras. For example, the measure algebra M(G),for a locally compact group G. Note that M(G) is amenable if and only if the group Gis amenable and discrete [4]. So, the amenability of M(G) forces to G to be discrete.Runde in [11] showed that the appropriate notion of amenability for M(G) is Connes-amenability, so that M(G) is Connes-amenable if and only if G is amenable.

The class of biprojective Banach algebras was introduced in [6] and [7]. Thisnotion is related to the notion of amenability, and since the appropriate notion ofamenability for dual Banch algebras is Connes-amenability, we have to define anappropriate analogue for biprojectivity of dual Banach algebras. For example, M(G)is Connes-biprojective if and only if G is amenable, but M(G) is biprojective if andonly if G is finite.

This work is devoted to the study of the structure of this new notion, which is calledConnes-biprojectivity. Such a study is important for obtaining homological results,and also, has an independent interest. The key result of this paper is that, ifA is a dualBanach algebra with a bounded approximate identity, then A is Connes-amenable ifand only ifA is Connes-biprojective (Theorem 2).

∗ speaker

215

A. Shirinkalam and A. Pourabbas

Let A be a dual Banach algebra. For a given A-bimodule E, let σWC(E) denotethe closed submodule of E of all elements x such that the mappingsA → E; a 7→ a · xand a 7→ x · a are σ(A,A∗) − σ(E, E∗)-continuous. Then A∗ ⊆ σWC((A⊗A)∗),from which it follows that π∗ maps A∗ into σWC((A⊗A)∗). Hence, π∗∗ drops to anA-bimodule homomorphism πσWC : σWC((A⊗A)∗)∗ → A.

2. Connes-biprojective dual Banach algebrasLet A be a dual Banach algebra. Then A is called Connes-biprojective if there

exists a bounded A-bimodule homomorphism ρ : A → σWC((A⊗A)∗)∗ such thatπσWC ρ = idA (that is, πσWC is a retraction).

In the following theorem, we determine the relation between Connes-biprojectivityand Connes-amenability. For a dual Banach algebraA, the following are equivalent :(i) A is Connes-amenable,(ii) A is Connes-biprojective and has a bounded approximate identity.

Example 2.1. Let A be a biflat dual Banach algebra. Then π∗ : A∗ → (A⊗A)∗

is a co-retraction and so π∗|A∗ : A∗ → σWC((A⊗A)∗) is a co-retraction again.This means that πσWC : (σWC(A⊗A)∗)∗ → A is a retraction, that is, A is Connes-biprojective.

By the previous example, every biprojective dual Banach algebra is Connes-biprojective. In the following example we see that the converse is false in general.

Example 2.2. Let G be a non-discrete amenable locally compact group. Then by [13,Theorem 4.4.13], M(G), the measure algebra of G, is a Connes-amenable dual Banachalgebra and thus by Theorem 2, M(G) is Connes-biprojective. Since G is not discrete,by [4, Theorem 1.3] M(G) is not amenable, so it is not biflat.

Clearly, every Connes-amenable Banach algebra is Connes-biprojective. Herewe give two examples of Connes-biprojective dual Banach algebras, which are notConnes-amenable.

Example 2.3. Let S be a discrete semigroup and let ℓ1(S ) be its semigroup algebra.Let A = ℓ1(S )∗. If ϕ is a character on c0(S ), then there exists a unique extension of ϕon c0(S )∗∗, (which is denoted by ϕ) and defined by

ϕ(F) = F(ϕ) (F ∈ c0(S )∗∗ A).

ϕ is a multiplicative map because, for every F,G ∈ A,

ϕ(FG) = (FG)(ϕ) = F(ϕ)G(ϕ).

Now we define a new multiplication onA by

ab = ϕ(a)b (a, b ∈ A, ϕ ∈ c0(S )∗).

With this multiplication,A becomes a Banach algebra which is a dual Banach space.We denote this algebra byAϕ. ThenAϕ is a biprojective Banach algebra. If we showthatAϕ is a dual Banach algebra, then by Example 2.1Aϕ is Connes-biprojective butAϕ is not Connes-amenable.

216


Example 2.4. Consider A =(0 C0 C

)with usual matrix multiplication and L1-norm.

Since A is finite dimensional, it is a dual Banach algebra. Clearly A has a rightidentity but it does not have an identity, so it is not Connes-amenable. But it is Connes-biprojective.

The next theorem shows that how Connes-biprojectivity deals with homomor-phisms.

Suppose that A is a Banach algebra, and that B is a dual Banach algebra. Letθ : A → B be a continuous homomorphism.

(i) Let θ∗ : B∗ → A∗ be such that θ∗|B∗ : B∗ → A∗ is surjective. Suppose that theimage of the closed unit ball of A is weak* dense in the closed unit ball of B.Then, biprojectivity ofA implies Connes-biprojectivity of B.

(ii) If A is dual and Connes-biprojective and θ is weak* continuous with weak*dense range, then B is Connes-biprojective.

Proof. Since θ : A → B is a continuous homomorphism, by [9, Proposition 1.10.10],we have the bounded linear map θ⊗θ : A⊗A → B⊗B. Also, consider the canonicalmap ı : B⊗B → σWC((B⊗B)∗)∗, which is norm continuous with weak* dense range.

(i) Suppose that A is biprojective. Then there is an A-bimodule homomorphismρA : A → A⊗A such that πA ρA = idA. Define a map ζ : A → σWC((B⊗B)∗)∗ byζ(a) = (ı (θ⊗θ) ρA)(a). Then ζ is bounded, and for every a, a′ ∈ A we have

ζ(aa′) = θ(a) • ζ(a′) = ζ(a) • θ(a′). (1)

For every b ∈ B, by assumption, there exists a bounded net (aα) ⊆ A such thatθ(aα) → b in the weak* topology. Since ζ is bounded, (ζ(aα))α has a weak* ac-cumulation point by Banach-Alaoglu theorem. Passing to a subnet (if it is neces-sary), w∗ − limα ζ(aα) exists. Thus, we can extend ζ to a weak* continuous mapρB : B → σWC((B⊗B)∗)∗ defined by ρB(b) = w∗ − limα ζ(aα). We need to verify thatρB is well-defined. In order to do this, it is enough to show that w∗ − limα ζ(aα) = 0 inσWC((B⊗B)∗)∗, whenever w∗ − limα θ(aα) = 0 in B. If λ ∈ A∗, then by assumption,there is a φ ∈ B∗ such that θ∗|B∗(φ) = λ. Now we have

limα⟨λ, aα⟩ = lim

α⟨θ∗(φ), aα⟩ = lim

α⟨φ, θ(aα)⟩ = 0.

Hence aα → 0 in the weak topology of A. Since ζ is weak-weak* continuous, weconclude that ζ(aα)→ 0 in the weak* topology.

Suppose that b and b′ ∈ B. Then there exist two nets (aα) and (a′β) in A such thatθ(aα) → b and θ(a′β) → b′ in the weak* topology. By the equation (1), and by theweak* continuity of the action of B we have, ρB(bb′) = w∗ − limα w∗ − limβ ζ(aαa′β) =w∗ − limα w∗ − limβ θ(aα) • ζ(a′β) = b • ρB(b′). Similarly, ρB(bb′) = ρB(b) • b′. ThusρB is a B-bimodule homomorphism.

217

A. Shirinkalam and A. Pourabbas

Finally, we prove that πBσWC ρB(b) = b for every b ∈ B. Observe that for theelementary tensor element a ⊗ a′ ∈ A⊗A we have

πBσWC ı (θ⊗θ)(a ⊗ a′) = πBσWC ı(θ(a) ⊗ θ(a′)) = θ(aa′) = θ πA(a ⊗ a′).

Thus for every a ∈ A,

πBσWC ζ(a) = θ(a). (2)

Now let b ∈ B and take a net (aα) ⊆ A such that θ(aα) → b in the weak*-topology.Then, by the equation (2), we have b = w∗ − limα θ(aα) = w∗ − limα π

BσWC ζ(aα) =

πBσWC(w∗ − limα ζ(aα)) = πBσWC ρB(b).(ii) Suppose thatA is dual and Connes-biprojective. Then there is anA-bimodule

homomorphism ρA : A → σWC((A⊗A)∗)∗ such that πAσWC ρA = idA. Considerthe map (θ⊗θ)∗ : (B⊗B)∗ → (A⊗A)∗ which is an A-bimodule map. We concludethat (θ⊗θ)∗ maps σWC(B⊗B)∗ into σWC(A⊗A)∗. Consequently, we obtain a weak*continuous map

φ := ((θ⊗θ)∗|σWC(B⊗B)∗)∗ : σWC((A⊗A)∗)∗ → σWC((B⊗B)∗)∗.

Thus by linearity and continuity and by the hypothesis, for every a ∈ A, we have

πBσWC φ ρA(a) = θ πAσWC ρA(a) = θ(a). (3)

Since the range of θ is dense, by (3) we have πBσWC ρB(b) = b for every b ∈ B.Hence B is Connes-biprojective.

It is well-known that ifA is an Arens regular Banach algebra, thenA∗∗, the bidualof A, is a dual Banach algebra with predual A∗(see [12] for more details). Supposethat A is an Arens regular Banach algebra. If A is biprojective, then A∗∗ is Connes-biprojective.

Let A be a dual Banach algebra and let I be a weak*-closed ideal of A. Then I isa dual Banach algebra with predual I∗ = A∗/I⊥. To see this, we have

(I∗)∗ = (A∗/I⊥)∗ = (I⊥)⊥ = I.

Since the multiplication in A is separately weak* continuous, a simple verificationshows that the multiplication on A/I is separately weak* continuous, so A/I is alsoa dual Banach algebra. If A is a Connes-biprojective dual Banach algebra and if I isa weak*-closed ideal of A which is essential as a left Banach A-module, then A/I isConnes-biprojective.

[1] J. W. Bunce and W. L. Paschke, Quasi-expectations and amenable von Neumann algebras, Proc.Amer. Math. Soc. 71 (1978), 232-236.

[2] A. Connes, Classification of injective factors, Ann. Math. 104 (1976), 73-114.[3] A. Connes, On the cohomology of operator algebras, J. Func. Anal. 28 (1978), 248-253.

218


[4] H. G. Dales, F. G hahramani and A. Ya. Helemskii, Amenability of measure algebras, J. LondonMath. Soc. 66 (2002), 213-226.

[5] A. Ya. Helemskii, The Homology of Banach and Topological Algebras, Kluwer Acad. Pub. vol.41, Dordrecht 1989.

[6] A. Ya. Helemskii, On a method for calculating and estimating the global homological dimensionsof Banach algebras, Mat. Sb. 87 (129) (1972), 122-135; English translation in Math. USSR Sb.16 (1972).

[7] A. Ya. Helemskii, The global dimension of a functional Banach algebra is different from one,Funkcional. Anal, i Prilozen. 6 (1972), no. 2, 95-96; English translation in Functional Anal. Appl.6 (1972).

[8] B. E. Johnson, R. V. Kadison and J. Ringrose, Cohomology of operator algebras, III, Bull. Soc.Math. France 100 (1972), 73-96.

[9] T. W. Palmer, Banach Algebras and the General Theory of ∗-Algebras, Vol. I, CambridgeUniversity Press, 1994.

[10] V. Runde, Amenability for dual Banach algebras, Studia Math. 148 (2001), 47-66.[11] V. Runde, Connes-amenability and normal, virtual diagonals for measure algebras, I, J. London

Math. Soc. 67 (2003), 643-656.[12] V. Runde, Dual Banach algebras: Connes-amenability, normal, virtual diagonals, and injectivity

of the predual bimodule, Math. Scand. 95 (2004), 124-144.[13] V. Runde, Lectures on Amenability, Lecture Notes in Mathematics, Vol. 1774, Springer-Verlag,

Berlin, 2002.

A. Shirinkalam,Faculty of Mathematics and Computer Science,Amirkabir University of Technology,Tehran 15914, Iran.e-mail: [email protected]

A. Pourabbas,Faculty of Mathematics and Computer Science,Amirkabir University of Technology,Tehran 15914, Iran.e-mail: [email protected]

219



January 20–21, 2016

Approximate amenability of semigroup algebra modulo an ideal

A. Soltani∗ and H. Rahimi

Abstract

In this talk, we consider approximately amenable Banach algebra modulo an ideal and present somecharacterization of this concept. Then using the obtained result, we investigate approximate amenabilityof semigroup algebras.

2010 Mathematics subject classification: Primary 99X99, Secondary 99Y99.Keywords and phrases: Amenability modulo an ideal, Approximate amenability modulo an ideal,Approximately inner, group congruence.

1. Introduction

The notion of approximately amenable of Banach algebra was introduced by Ghahra-mani and Loy in [2]. The structure of approximately amenable (contractible) Banachalgebras is considered in [2] through different ways. It is shown two concepts of ap-proximately amenable and approximately contractible for Banach algebras are equiv-alent and for a locally compact group G, the group algebra L1(G) is approximatelyamenable (contractible) if and only if G is amenable. For a discrete semigroup S , thenecessary and sufficient conditions for approximately amenable (contractibility) of thesemigroup algebra l1(S ) is quite more complicated. It has not been explored whenl1(S ) is approximately amenable. For discrete semigroup S , it has been shown thatif l1(S ) is approximately amenable, then S is regular and amenable [3], and if S isright cancellative semigroup such that l1(S ) is approximately amenable then S is anamenable group and l1(S ) is amenable [7]. To see some remarks of approximatelyamenable see [3, 5, 7].

The concept of amenability modulo an ideal for the l1- semigroup algebra wasintroduced by M. Amini and the second author in [1]. The effect is to gather up thenon-group structure into a congruence and its corresponding ideal and establish someanalogue of Johnson’s theorem on groups and their l1-algebras.∗ speaker∗ speaker

8

220

A. Soltani and H. Rahimi

2. PREMINARIES

Let A be Banach algebra and I be a closed ideal of I. A is called amenable modulo Iif every bounded derivation D : A→ X∗ (D : A→ X) is inner on A\I := a ∈ A : a < Ifor all Banach A-bimodule X such that I · X = X · I = 0. By a virtual diagonal moduloI we mean an element M ∈ ( A

I ⊗A)∗∗ such that a · π∗∗AI− a = 0 (a ∈ A, a ∈ A

I ) anda · M − M · a = 0 (a ∈ A \ I), an approximate diagonal modulo I we mean a boundednet (mα)α ⊂ ( A

I ⊗A) such that a·π AImα−a→ 0 (a ∈ A) and a·mα−mα ·a→ 0 (a ∈ A\I).

It was shown that a Banach algebra A is amenable modulo I if and only if there is anapproximate diagonal modulo I, if and only if there is a virtual diagonal modulo I.

3. Approximately amenable Banach algebras modulo an ideal

Definition 3.1. Let A be a Banach algebra and I be a closed ideal of A. A is calledapproximate amenability modulo I, if every bounded derivation D : A → X∗ thereexists net (xα)α ⊆ X∗, such that D(a) = limα adxα(a) on the set theoretic difference A\ Ifor all Banach A-bimodule X such that I · X = X · I = 0

Theorem 3.2. Let I be a closed ideal of A. Then the following assertions holds;(i) if A

I is approximately amenab and I2 = I then A is approximate amenabilitymodulo I;

(ii) if A is approximate amenability modulo I, then AI is approximately amenable.

(iii) If A is approximate amenability modulo I and I is approximately amenable,then A is approximately amenable.

We racall that for a normes algebra A, the unitization of A over

Theorem 3.3. A is approximate amenability modulo I if and only if A♯ is approximateamenability modulo I.

Theorem 3.4. A is approximate amenability modulo I if and only if either of thefollowing equivalent conditions hold:

(a) There is a net (Mν) ⊂ ( A♯I ⊗A♯)∗∗ such that a ·Mν−Mν ·a→ 0 and π∗∗(Mν) −→ e

(∀a ∈ A♯ \ I);(b) There is a net (M′ν) ⊂ ( A♯

I ⊗A♯)∗∗ such that a ·M′ν −M′ν ·a −→ 0 and π∗∗(M′ν) = efor every ν (∀a ∈ A♯ \ I).

4. Approximate amenability modulo an ideal of semigroup algebras

Let S be a semigroup and E = E(S ) be the set (possibly empty) of idempotentsof S . We recall that a semigroup S is called an E-semigroup if E(S ) forms a sub-semigroup of S , E-inversive if for all x ∈ S , there exists y ∈ S such that xy ∈ E(S ),regular if V(a) , ϕ where V(a) = x ∈ S : a = axa, x = xax is the set of inverses ofa ∈ S and S is called an inverse semigroup if moreover, the inverse of each element isunique. An inverse semigroup S is called E-unitary if for each x ∈ S and e ∈ E(S ),ex ∈ E(S ) implies x ∈ E(S ) and S is called semilattice if S is a commutative and

221

Approximate amenability modulo an ideal

idempotent semigroup. Also, a semigroup S is called eventually inverse if everyelement of S has some power that is regular and E(S ) is a semilattice.

A congruence ρ on semigroup S is called group congruence if S/ρ is group. Wedenote the least group congruence on semigroup S by σ. Following Gigon [4], if Sis an E-inversive E-semigroup with commuting idempotents or S is an eventuallyinverse semigroup then σ = (a, b) ∈ S × S | ea = f b for some e, f ∈ E(S ) is theleast group congruence on S . The kernel of a congruence ρ on a semigroup S is the seta ∈ S : aρ ∈ E(S/ρ) = a ∈ S : (a, a2) ∈ ρ. We now recall the following result of[1].

Theorem 4.1. Let S be a semigroup. If S is to one of the following statements;

(i) S is an E-inversive semigroup with commuting idempotents;(ii) S be an eventually inverse semigroup;

Then S is amenable if and only if l1(S ) is approximate amenability modulo Iσ.

Lemma 4.2. The following statements hold:(i) if S is a semigroup, ρ is a congruence on S and ω is a weight on S , thenl1(S ,ω)

Iρ≃ l1(S/ρ, ωρ) where ωρ([s]ρ) = infω(s) : s ∈ [s]ρ is the induced weight on

S/ρ;(ii) if S is an E-inversive semigroup with commuting idempotents or S is aneventually inverse semigroup, σ be the least group congruence on S and ω be aweight on S , then l1(S/σ, ωσ)) ≃ l1(S ,ω)

Iσwhere Iσ is a closed ideal of l1(S , ω) and

I2σ = Iσ.

Theorem 4.3. Suppose that ω is a weight on semigroup S . If S is to one of thefollowing statements:


Then S is amenable and Ωωσ is bounded if and only if l1(S , ω) is approximateamenability modulo Iσ.

By a similar argument of Theorem 4.4, we have the following result;

Corollary 4.4. Suppose that ω is a weight on semigroup S . If S is to one of thefollowing statements:


Then S is amenable and Ωωσ is bounded if and only if l1(S , ω) is amenable modulo Iσ.

222

A. Soltani and H. Rahimi

[1] M. Amini, H. Rahimi, Amenability of semigroups and their algebras modulo a group congruence,Acta Mathematica Hungarica vol 144, Issue 2,(2014), 407-415.

[2] F. Ghahramani and R. J. Loy, Generalized notions of amenabilit, J. Func. Analvol 208 (2004),229-260.

[3] F. Ghahramani, R. J. Loy, and Y. Zhang, Generalized notions of amenability, II,J. FunctionalAnalysisvol 254 (2008), 1776-1810.

[4] R. S. Gigon, Congruences and group congruences on a semigroup,Semigroup Forumvol 86 (2013),431-450.

[5] H. M. Ghlaio and C. J. Read, Irregular commutative semigroups S with weakly amenablesemigroup algebra 11(S ),Semigroup Forum vol 82, Issue 2 (2011), 367-383.

[6] H. Rahimi and E. Tahmasebi, Amenability and Contractibility modulo an ideal of Banach algebras,Abstract and Applied Analysis.

[7] N. Gronbaek, Amenability of weighted discrete convolution algebras on cancellative semi-groups,Proc. Roy. Soc. Edinburgh, Section A vol 110 (1988), 351-360.

A. Soltani,Department of Mathematics, Faculty of Science,University of Central Tehran Branch, Islamic Azad University,City Tehran, Irane-mail: [email protected]

H. Rahimi,Department of Mathematics, Faculty of Science,University of Central Tehran Branch, Islamic Azad University, P. O. Box 13185/768,City Tehran, Irane-mail: [email protected]

223



January 20–21, 2016

The weighted KPC-hypergroups

S. M. Tabatabaie and F. Haghighifar∗

AbstractIn this paper, we extend some basic concepts and results from weighted DJS–hypergroups to KPC–hypergroups. Also, we define a new convolution product on the space L1(Q,w).

2010 Mathematics subject classification: 28C15, 20N20.Keywords and phrases: DJS-hypergroup, Haar measure, KPC-hypergroup, weight function, weightedhypergroups.

1. Introduction and PreliminariesRoughly speaking, a hypergroup is a topological space equipped with an extra struc-ture, which leads to the construction of a Banach algebra on the Banach space of allbounded complex Radon measures on the hypergroup. Locally compact hypergroups,as an extension of locally compact groups, were introduced in a series of papers byDunkl [4], Jewett [8], and Spector [12] in 70’s (we refer to this definition of hyper-group as DJS–hypergroup). For more details about DJS–hypergroups we refer to [1].Kalyuzhnyi, Podkolzin, and Chapovsky [9] introduced new axioms for hypergroups in2010. This new concept is an extension of DJS–hypergroups, and also generalizes anormal hypercomplex system with a basis unity to the nonunimodular case. We refer tothis notion as KPC–hypergroup. They studied harmonic analysis on KPC–hypergroupsand showed that there is an example of a compact KPC–hypergroup related to thegeneralized Tchebycheff polynomials, which is not a DJS–hypergroup [9].

Some topics which are related to hypergroups and have been initiated based on asimilar study on locally compact groups are "weighted hypergroups" and "weightedhypergroup algebras". The first studies on weighted hypergroup algebras may betracked back to [2], [5], and [6]. The weighted space L1(G,w) of a locally compactgroup G was studied extensively ([3], [10], [11]). The basic idea is to consider acontinuous weight w and change the norm of L1(G) by a factor of w. The new spaceL1(G,w) then consists of those Borel measurable functions on G, for which f w is inL1(G).

Let Q be a locally compact Hausdorff space. We denote by M(Q) the space ofall complex Radon measures on Q, and by C(Q), the space of all complex-valuedcontinuous functions on Q. The support of a function f is denoted by supp( f ).

224

S.M. Tabatabaie, F. Haghighifar

In the sequel, we recall the definition and some basic properties of the locallycompact cocommutative KPC–hypergroups. For more details we refer to [9].

Definition 1.1. Let Q be a locally compact second countable Hausdorff space with aninvolutive homeomorphism ⋆ : Q −→ Q satisfying the following conditions:1. there is an element e ∈ Q such that e⋆ = e;2. there is a C-linear mapping ∆ : C(Q)→ C(Q × Q) such that

i. ∆ is co-associative, that is,

(∆ × id) ∆ = (id × ∆) ∆;

ii. ∆ is positive, that is, ∆ f ≥ 0 for all f ∈ C(Q) such that f ≥ 0;iii. ∆ preserves the identity, that is, (∆1)(p, q) = 1 for all p, q ∈ Q;iv. For all f , g ∈ Cc(Q) we have (1 ⊗ f ).(∆g) ∈ Cc(Q × Q) and ( f ⊗ 1).(∆g) ∈

Cc(Q × Q).3. the homomorphism ϵ : C(Q) → C defined by ϵ( f ) = f (e), satisfies the counit

property, that is,(ϵ × id) ∆ = (id × ϵ) ∆ = id,

in other words, (∆ f )(e, p) = (∆ f )(p, e) = f (p) for all p ∈ Q.4. the function f defined by f (q) = f (q⋆) for f ∈ C(Q) satisfies

(∆ f )(p, q) = (∆ f )(q⋆, p⋆).

5. there exists a positive measure m on Q, supp m = Q, such that∫Q

(∆ f )(p, q)g(q)dm(q) =∫

Qf (q)(∆g)(p⋆, q)dm(q)

for all f ∈ Cb(Q) and g ∈ Cc(Q), or f ∈ Cc(Q) and g ∈ Cb(Q), p ∈ Q; such ameasure m will be called a left Haar measure on Q.

Then (Q, ⋆, e,∆,m), or simply Q, is called a locally compact KPC–hypergroup.

Throughout this paper, Q is a locally compact KPC–hypergroup and m is a leftHaar measure on Q.Notation. In the above definition we have used the following notations:

[(∆ × id) ∆( f )](p, q, r) := ∆(∆ f (p, ·))(q, r),

[(id × ∆) ∆( f )](p, q, r) := ∆(∆ f (·, q))(p, r),

[(ϵ × id) ∆( f )](p) := ϵ(∆ f (p, ·)) = ∆ f (p, e),

[(id × ϵ) ∆( f )](p) := ϵ(∆ f (·, p)) = ∆ f (e, p),

( f ⊗ 1)(p, q) · (∆g)(p, q) = f (p)1(q) · ∆g(p, q),

(1 ⊗ f )(p, q) · (∆g)(p, q) := 1(p) f (q) · ∆g(p, q),

where f ∈ C(Q) and p, q, r ∈ Q.

225

The Weighted KPC-Hypergroups

Definition 1.2. Let µ, ν ∈ M(Q) be such that the linear functional µ ∗ ν defined by

(µ ∗ ν)( f ) =∫

Q

∫Q∆( f )(p, q)dµ(p)dν(q), ( f ∈ C(Q))

is a measure. Then the measures µ and ν are called convolvable. Specially, we have(δp ∗ δq)( f ) = (∆ f )(p, q), where p, q ∈ Q.

If µ, ν ∈ M(Q) are bounded, then µ and ν are convolvable ([9], Lemma 3.3).

Definition 1.3. Let m be a left Haar measure on Q. The convolution of complex-valuedBorel measurable functions f and g on Q is denoted by f ∗ g and is defined by

( f ∗ g)(q) =∫

Qf (p)(∆g)(p⋆, q)dm(p),

where q ∈ Q.

2. The Weight Functions on KPC–Hypergroups

Definition 2.1. A complex-valued bounded continuous function k on Q is calledpositive definite if for any n ∈ N, and qi ∈ Q (i = 1, ..., n), the matrix

((∆k)(q⋆i , q j))1≤i, j≤n

is positive semi-definite [7].

Definition 2.2. A continuous function w : Q → [0,∞) is called a weight function, ifw(e) = 1 and

∆w(p, q) ≤ w(p)w(q) (p, q ∈ Q).

Throughout this paper, w is a symmetric weight function on Q, i.e. w(p⋆) = w(p), forall p ∈ Q.

Definition 2.3. A complex-valued function f on Q is called w–bounded, if there is aconstant K > 0 such that for all p ∈ Q, | f (p)| ≤ Kw(p).

Lemma 2.4. Let f be a positive definite function on Q. Then for all p ∈ Q we have

i. f (p⋆) = f (p), where f is the complex conjugate of f ;ii. f (e) ≥ 0;iii. ∆ f (p, p⋆) ≥ 0;iv. | f (p)|2 ≤ ∆ f (p, p⋆) f (e);v. | f (p)| ≤ 1

2 (∆ f (p, p⋆) + f (e)).

Remark 2.5. Easily, we can see that for each f ∈ C(Q) and p, q ∈ Q, |∆ f (p, q)| ≤∆| f |(p, q).

Lemma 2.6. If f is a w–bounded function on Q, then there is a constant K > 0 suchthat |∆ f | ≤ K∆w.

226

S.M. Tabatabaie, F. Haghighifar

Proposition 2.7. If f is a w–bounded positive definite function, then for all p ∈ Q,| f (p)| ≤ f (e)w(p).

Proposition 2.8. Let w be a weight function on Q. Then A defined by

A(µ) :=∫

Qwd|µ|

is submultiplicative on Mc(Q).

Notation. Let Q be a cocommutative KPC–hypergroup with a Haar measure m,and w be a weight function on Q. We denote the set of all complex-valued Borelmeasurable functions f on Q such that

∫Q | f |w dm < ∞, by L1

w(Q,m). To abbreviation,we write L1

w(Q) instead of L1w(Q,m).

Theorem 2.9. Let w(p) ≥ 1 for all p ∈ Q. For any f ∈ L1w(Q) we define ∥ f ∥1,w :=∫

Q | f |wdm. Then (L1w(Q), ∥.∥1,w, ∗) is a normed subalgebra of L1(Q).

Proposition 2.10. Let f ∈ L1w(Q). Then ∥ fp∥1,w ≤ w(p)∥ f ∥1,w, where p ∈ Q and

fp(q) = ∆ f (p, q).

Lemma 2.11. If w > 0 is a weight on Q, then Cc(Q) is dense in L1w(Q).

Remark 2.12. The dual space of Llw(Q) is L∞w (Q), formed by all complex-valued

measurable functions ϕ on Q such that ϕw ∈ L∞(Q) (with the usual convention that

two such functions are same if they coincide locally almost everywhere), and we define

∥.∥∞,w = ess.supp∈Q|ϕ(p)|w(p)

.

That is, the bounded linear functionals on L1w(Q) are precisely those of the form

f 7→ ⟨ f , ϕ⟩, where

⟨ f , ϕ⟩ =∫

Qf (p)ϕ(p)dm(p) ( f ∈ L1

w(Q), ϕ ∈ L∞w (Q)),

with the norm ∥ϕ∥∞,w. If ϕ ∈ L∞w (Q) is continuous, then for all p ∈ Q, |ϕ(p)| ≤∥ϕ∥∞,ww(p).

[1] W. R. Bloom and H. Heyer, An Harmonic Analysis of Probability Measures on Hypergroups, DeGruyter Studies in Mathematics, Walter de Gruyter, Berlin, Germany, 1995.

[2] W. R. Bloom and P. Ressel, Expontially bounded positive definite functions on a commutativehypergroup .Math. Soc. (Series A) 61 (1996), 238–248.

[3] H. G. Dales and A. T. M. Lau, The second duals of Beurling algebras, Memoirs Amer. Math.Soc., 117, American Mathematical Society, Providence, R.I., 2005.

[4] C. F. Dunkl, The measure algebra of a locally compact hypergroup, Trans. Amer. Math. Soc. 179(1973), 331–348.

[5] F. Ghahramani and A. R. Medghalchi, Compact multipliers on weighted hypergroup algebras,Math. Proc. Cambridge Philos. Soc., 98 (1985), 493–500.

227

The Weighted KPC-Hypergroups

[6] F. Ghahramani and A. R. Medghalchi, Compact multipliers on weighted hypergroup algebras II,Math. Proc. Cambridge Philos. Soc., 100 (1986), 145–149.

[7] R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 1985.[8] R. I. Jewett, Spaces with an abstract convolution of measures, Advances in Mathematics, 18

(1975), 1–101.[9] A. A. Kalyuzhnyi, G. B. Podkolzin, and Yu. A. Chapovski, Harmonic Analysis on a Locally

Compact Hypergroup, Methods of Functional Analysis and Topology, 16 (2010), 304–332.[10] T. W. Palmer, Banach Algebras and the General Theory of ∗–Algebras II, Cambridge University

Press, Cambridge, 2001.[11] H. Reiter and J. D. Stegeman, Classical Harmonic Analysis and Locally Compact Groups, 2nd

ed., London Mathematical Society Monographs, New Series 22, Clarendon Press, Oxford, 2000.[12] R. Spector, Measures Invariant Sure Less Hypergroups, Trans. Amer. Math. Soc. 239 (1978),

147–165.

S. M. Tabatabaie,Department of Mathematics,University of Qom,Qom, Iran.e-mail: [email protected]

F. Haghighifar∗,Department of Mathematics,University of Qom,Qom, Iran.e-mail: [email protected]

228



January 20–21, 2016

A dense additive subsemigroup of L2(G)

S. M. Tabatabaie

Abstract

In this paper we study some conditions for an additive semigroup generated by a set M ⊆ L2(G) to bedense in L2(G).

2010 Mathematics subject classification: Primary 41A65, Secondary 22A10, 46B20, 46B25.Keywords and phrases: Locally compact group, Banach space, Approximation, Uniformly smooth space..

1. Introduction and Notations

Let X be a Banach space, M ⊆ X, and R(M) be the additive subsemigroup of Xgenerated by M, that is

R(M) :=Σn

j=1x j : n ∈ N, x j ∈ M.

Finding necessary or sufficient conditions on M or X for the set R(M) to be densein X is subject of many papers. See [1] and [5] . In [4] the set R(M) is called theadditive resultant of M or the semimodule generated by M and is denoted by (M)a.Since this problem involves no multiplication by complex scalars, the space X maybe assumed to be real. In this work we study this question for the case X = L2(G),where G is a locally compact abelian group, and we give an extension of the Wiener’sclassical theorem [6] on locally compact abelian groups, which is very useful in theproof of our main result. We follow from [3] for related notations.

2. Main Results

Theorem 2.1. Let G be a locally compact abelian group, ϵ > 0, and A :=Σn

j=1λ j⟨x j, ·⟩ : n ∈ N, λ j ∈ C, x j ∈ G, where for each ξ ∈ G, ⟨x j, ξ⟩ = ξ(x j). Iff ∈ L2(G) has a null zeros set, then for each g ∈ L2(G) there exists an ψ ∈ A such that

∥g − ψ f ∥2 < ϵ.Definition 2.2. A Banach space X with unit sphere S (X) is said to be uniformly smoothif for every ϵ > 0 there is a δ > 0 such that whenever x ∈ S (X) and ∥y∥ < δ then∥x + y∥ + ∥x − y∥ ≤ 2 + ϵ∥y∥.

229

S. M. Tabatabaie

Remark 2.3. A Banach space X is uniformly smooth if and only if its modulus ofsmoothness

s(τ) := sup∥∥∥∥∥ x + y

2

∥∥∥∥∥ + ∥∥∥∥∥ x − y2

∥∥∥∥∥ − 1 : ∥x∥ = 1, ∥y∥ = τ, τ ≥ 0,

satisfies s(τ) = o(τ) as τ→ 0.[2]

Here we recall the following lemma from [1].

Lemma 2.4. Let F be a closed additive subgroup in a uniformly smooth Banach spaceX with modulus of smoothness s(τ), τ ≥ 0. Suppose that a, b ∈ F and for every ϵ > 0there are x0, x1, . . . , xn ∈ F such that x0 = a, xn = b and Σn

k=1s (∥xk − xk−1∥) < ϵ. Then[a, b] ⊆ F, where [a, b] = (1 − t)a + tb : 0 ≤ t ≤ 1 .

Definition 2.5. Suppose that X is a Banach space, G is a locally compact abeliangroup, U0 is a compact neighborhood of e in G, and ϕ : U0 → X is continuous. Thenthe modulus of continuity of ϕ is defined by

ωϕ(U) := sup∥ϕ(xy) − ϕ(x)∥ : y ∈ U, xy ∈ U0,

where U ⊆ U0 is a neighborhood of e in G.

Definition 2.6. Suppose that G is a locally compact abelian group, f ∈ L2(G), and Uis a compact neighborhood of e in G. Then the integral modulus of continuity of f isdefined by

ω2( f ,U) := supx∈U∥ fx−1 − f ∥2.

We denote ω2( f ,U) = o(U) as U → e if

limU→e

ω2( f ,U)m(U)

= 0.

We have the following results for some special locally compact abelian groups.

Theorem 2.7. Let X be a uniformly smooth space with modulus of smoothness s(τ), Fbe a closed additive subgroup of X, and ϕ : U0 → F be a continuous map satisfiess(ωϕ(U)) = o(U) as U → e, where U0 is a compact neighborhood of e. Then Fcontains the closed R-vector subspace L spanned by elements of the form a− b, wherea, b ∈ ϕ(U0).

Corollary 2.8. Suppose that f ∈ L2(G), π(ξ ∈ G : f (ξ) = 0) = 0, and theintegral modulus of continuity of f satisfies ω2( f ,U) = o(U) as U → e. ThenR(± fx : x ∈ G) is dense in L2(G).

230

A dense additive subsemigroup of L2(G)

[1] P. A. Borodin, Density of a semigroup in a Banach space, Izvestiya: Mathematics 78:6 (2014)1079-1104.

[2] J. Diestel, Geometry of Banach Spaces-selected topics, Lecture notes in Math. vol. 485, Springer-Verlag, Berlin-New York, 1975.

[3] G. B. Folland, A Course in Abstract Harmonic Analysis, CRC Press, London, 1995.[4] E. Hille and R. S. Phillips, Functional analysis and semigroups, Amer. Math. Soc. Colloq. Publ.

31 (1957).[5] V. Temlyakov, Nonlinear Methods of approximation, Found. Comput. Math. 3:1 (2003) 33-107.[6] N. Wiener, The Fourier Integral and Certain of Its Applications, Univ. Press, Cambridge, 1933.

S. M. Tabatabaie,Department of Mathematics,University of Qom,Qom, Irane-mail: [email protected]

231



January 20–21, 2016

Paradoxical decomposition of hypergroups

A. Yousofzadeh

Abstract

The paradoxical decomposition of a discrete hypergroup is defined. It is shown that a discrete hypergroupis amenable if and only if it admits no paradoxical decomposition.

2010 Mathematics subject classification: Primary 43A62 , Secondary 43A07.Keywords and phrases: Hypergroup; paradoxical decomposition; amenability .

1. Introduction

A hypergroup H is a locally compact Hausdorff topological space for which there is abinary operation ∗, called convolution, on the vector space of bounded Radon measuresturning it into an algebra such that for x, y ∈ H, the convolution of the two pointmeasures is a probability measure, with compact support and the map (x, y) 7→ δx ∗ δy

is continuous. Also the map (x, y) 7→ supp(δx ∗ δy) is continuous with respect tothe Michael topology on the space of compact subsets of H. It is also supposed thatthere is a unique element e ∈ H such that for every x ∈ H, δx ∗ δe = δe ∗ δx = δx.For a locally compact hypergroup we assume that there exists a homeomorphismˇ : H → H such that for all x ∈ H, ˇx = x, which can be extended to M(H) viaµ(A) = µ(x ∈ H : x ∈ A), and such that (µ ∗ ν) = ν ∗ µ and finally for x, y ∈ H,e ∈ supp(δx ∗ δy) if and only if y = x. Using the concept of configurations, B.Willson obtained an equivalent relation for the amenability of hypergroups. We usethis concept to introduce and construct the paradoxical decomposition of hypergroups.

2. Preliminaries

Let H be a hypergroup and E = E1, . . . , Em be a finite measurable partition of H.Choose an n-tuple of elements of H, h = h1, . . . , hn. A configuration is an (n + 1)-tuple C = (C0,C1, . . . ,Cn) where each C j ∈ 1, . . . ,m. For a fixed configuration C,we define ξ0(C) to be the real-valued function on H given by:

ξ0(C)(x) := Πnj=0δh j ∗ δx(EC j ),

here we convent that h0 = e.

232

A. Yousofzadeh

Definition 2.1. Fix E and h as before. Let zC : C ∈ Con(h,E) be variablescorresponding to the m(n+1) configurations. Consider the m×n configuration equations∑

C∈Con(h,E),C0=i

zC =∑

C∈Con(h,E),C j=i

zC

for each i ∈ 1, . . . ,m and j ∈ 1, . . . , n. We say that a solution to these configurationequations is positive if, for each C ∈ Con(h,E), zC ≥ 0; normalized if

∑C∈Con(h,E) zC =

1; and inequality preserving if for every choice of m(n+1) real numbers αC : C ∈Con(h,E),

0 ≤∑

C∈Con(h,E)

αCξ0(C) a.e.⇒ 0 ≤∑

C∈Con(h,E)

αC zC .

Theorem 2.2. [2] Let H be a hypergroup with left Haar measure λ. H is amenable ifand only if for all choices of m, n, h and E the m × n configuration equations have apositive, normalized, inequality preserving solution.

Definition 2.3. C ⊆ Rn is a cone in Rn if for any vector y ∈ C and k > 0 we haveky ∈ C. The cone C is pointed if C ∩ (−C) = 0. In what follows C∗ will denote thepolar cone of an arbitrary cone C in Rn; that is

C∗ = y∗ ∈ Rn; (y∗)′y ≥ 0, f or all y ∈ C.

Theorem 2.4. [1] Let M be a non-vacuous m×n matrix and let C be a cone inRn whichis closed, convex and pointed. Then one and only one of the following statements isconsistent

1) Mx = 0 for some x ∈ C, x , 0,

2) M′y ∈ Int(−C∗), y ∈ Rm.

3. Main Theorem

Definition 3.1. Suppose that H is a discrete hypergroup. Let

F = f1, . . . , fn, g1, . . . , gm

be a finite family of nonnegative real valued bounded functions on H and h =h1, . . . , hn, h′1, . . . , h

′m be a subset of H. We say that (F , h) is a paradoxical decompo-

sition of H if

1)∑n

i=1 fi +∑m

j=1 g j = 1

2)∑n

i=1 δhi∗ fi = 1

3)∑m

j=1 δh′ j∗ g j = 1.

233

Paradoxical decomposition of hypergroups

Remark 3.2. Clearly if H is a group, then Definition 3.1 is the definition of a completeparadoxical decomposition of H as a group if the functions fis and g js are disjointlysupported.

Let H be amenable and µ be the left invariant mean on C(H). If H admits aparadoxical decomposition as in Definition 3.1, then we have

1 = µ(1) = µ(n∑

i=1

fi +m∑

j=1

g j)

= µ(n∑

i=1

δhi∗ fi) + µ(

m∑j=1

δh′ j∗ g j)

= µ(1) + µ(1) = 2,

which is impossible. Now assume that H is not amenable. By Theorem 2.2 there areh and E such that the system of configuration equations corresponding to (h,E) has nonormalized inequality preserving solution. Set

A =(zi) ∈ Rmn, zi ≥ 0, and (zi) is inequality preserving

.

Then A is obviously a pointed convex closed positive cone in Rmn. So by Theorem2.4 there exists y ∈ Rmn such that∑

C∈Con(h,E)

(M′y)Cξ0(C) > 0.

Now an application of the definition of configuration equations leads us to the follow-ing theorem

Theorem 3.3. Hypergroup H is amenable if and only if H admits no paradoxicaldecomposition.

In what follows let ∑C∈Con(h,E),C0=i

zC =∑

C∈Con(h,E),C j=i

zC

be the system of configuration equations corresponding to the pair (E, h) and X =(zC)C∈Con(h,E) be the vector of variables corresponding to m(n+1) configurations. Denotethe matrix form of this system by (B − A)X = 0. Clearly A and B are matrices withentries in 0.1.

If P =

P1P2...

Pn

is a permutation matrix, by P+ we mean the matrix with shifted

234

A. Yousofzadeh

rows, i.e.

P+ =

P2P3...

Pn

P1

,

Also we use the notation

T =

1 0 0 . . . 01 1 0 . . . 0...1 1 1 . . . 1

.Definition 3.4. A configuration equation (B − A)X = 0 is called normal if there existsa permutation matrix P such that the matrix

T P(B − A) − P+A (1)

has integer entries equal or greater than −1.

Theorem 3.5. If hypergroup H admits a normal configuration equation, then it isparadoxical, i.e. it is non-amenable. Moreover the paradoxical decomposition iscompletely constructed.

Acknowledgement

The author would like to thank the Islamic Azad University, Mobarakeh Branch.

[1] B. Skarpness, and V.A. Sposlto, A note on Gordan’s theorem over cone domains, Internat. J. h.Math. Sci. 5 No. 4 (1982) 809-812.

[2] B. Willson, Configurations and invariant nets for amenable hypergroups and related algebras,Trans. Amer. Math. Soc. 366 (2014), no. 10, 5087-5112.

A. Yousofzadeh,Department of Mathematics,Islamic Azad University, Mobarakeh Branch,Isfahan, Irane-mail: [email protected]

235

Proceedings of · The 4th Seminar on Harmonic Analysis and Applications Department of Mathematics,...

Documents

Transcript of Proceedings of · The 4th Seminar on Harmonic Analysis and Applications Department of Mathematics,...