Operato r Theory - American Mathematical Society · A rich theory of completely positive and...
Transcript of Operato r Theory - American Mathematical Society · A rich theory of completely positive and...
Operato r Theor y Operato r Algebra s
an d Application s
http://dx.doi.org/10.1090/pspum/051.2
PROCEEDING S O F SYMPOSI A IN PUR E MATHEMATIC S
Volum e 51 , Par t 2
Operato r Theor y Operato r Algebra s
an d Application s
Willia m B. Arveso n an d Ronal d G . Douglas , Editor s
AMERICA N MATHEMATICA L SOCIET Y
PROVIDENCE , RHOD E ISLAN D
PROCEEDINGS OF THE SUMMER RESEARCH INSTITUTE ON OPERATOR THEORY/OPERATOR ALGEBRAS AND APPLICATIONS
HELD AT UNIVERSITY OF NEW HAMPSHIRE DURHAM, NEW HAMPSHIRE
JULY 3-23, 1988
with support from the National Science Foundation, Grant DMS-8714162
1980 Mathematics Subject Classification (1985 Revision). Primary 46L, 47A, 47B, 58G.
Library of Congress Cataloging-in-Publieation Data
Operator theory: operator algebras and applications/William B. Arveson and Ronald G. Douglas, editors. p. cm.—(Proceedings of symposia in pure mathematics; v. 51) ISBN 0-8218-1486-9 I. Operator algebras — Congresses. 2. Operator theory — Congresses. I. Arveson, William. II. Douglas, Ronald G. III. Series. QA326.067 1990 90-33771 512/.55-dc20 CIP
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Contents Preface xi
PART 1
The spectral C* -algebra of an iso-semigroup WILLIAM ARVESON 1
Two-sided Lagrange-Sylvester interpolation problems for rational matrix functions J. A. BALL, I. C. GOHBERG, AND L. RODMAN 17
Nonstationary inertia theorems, dichotomy, and applications A. BEN-ARTZI AND I. GOHBERG 85
Toeplitz operators, quantum mechanics, and mean oscillation in the Bergman metric LEWIS COBURN 97
Towards a functional calculus for subnormal tuples: The minimal normal extension and approximation in several complex variables JOHN B. CONWAY 105
On the technique of comparison algebra for elliptic boundary problems on noncompact manifolds H. O. CORDES 113
Composition operators on Hilbert spaces of analytic functions: A status report CARL C. COWEN 131
Random Toeplitz operators R. E. CURTO, P. S. MUHLY, AND J. XlA 147
Isomorphisms of nest algebras and their quotients KENNETH R. DAVIDSON 171
Invariants for Hilbert modules RONALD G. DOUGLAS 179
Multivariate multipliers for groups and their operator algebras E. G. EFFROS AND Z.-J. RUAN 197
Beyond commutant lifting J. WILLIAM HELTON 219
VI CONTENTS
Similarity and approximation of operators DOMINGO A. HERRERO 225
A primer on A^-theory NIGEL D. HIGSON 239
Complete distributivity ALAN HOPENWASSER 285
Operator algebraic invariants for elliptic operators JERRY KAMINKER 307
The supersymmetric replica trick and smoothness of the density of states for random Schrodinger operators ABEL KLEIN 315
Some recent progress in nest algebras DAVID R. LARSON 333
Rigidity theorems in spaces of analytic functions VERN I. PAULSEN 347
Hankel operators and multivariate stationary processes VLADIMIR V. PELLER 357
The principal index JOEL PINCUS 373
Refinement theory for nonselfadjoint operator algebras STEPHEN C. POWER 395
Some remarks on the index theory for semigroups of endomorphisms of 93(55) ROBERT T. POWERS 405
Deformation quantization and operator algebras MARC A. RIEFFEL 411
Toeplitz and Hankel operators, wavelets, NWO sequences, and almost diagonalization of operators RICHARD ROCHBERG 425
K and KK : Topology and operator algebras JONATHAN ROSENBERG 445
Applications of C*-algebras and operator theory to proper holomorphic mappings NORBERTO SALINAS 481
Function theory and de Branges's spaces DONALD E. SARASON 495
Algebraic quantization and stability IRVING SEGAL 503
Analytic operator algebras BARUCH SOLEL 519
CONTENTS vii
Automorphisms which preserve unitary equivalence classes of normal states
U. HAAGERUP AND E. STORMER 531
Cocycle conjugacy of group actions on factors
MASAMICHI TAKESAKI 539
Pseudodifferential operators and ^-homology
MICHAEL E. TAYLOR 561
Toeplitz operators and index theory in several complex variables
HARALD UPMEIER 585
Szego expansions for operators with smooth or nonsmooth symbol
HAROLD WIDOM 599
Analytic theory of subnormal n-tuple of operators
DAOXING XIA 617
PART 2
Reduced Hankel operators and radial measures GREGORY T. ADAMS 1
From algebras of normal operators to intersecting hyperplanes
E. A. AZOFF AND H. A. SHEHADA 11
Random walks, AT0-theory for AF algebras, and classical statistical mechanics
B. M. BAKER AND D. E. HANDELMAN 17
Some C4 and C$ norm inequalities related to the paving problem
K. BERMAN, H. HALPERN, V. KAFTAL, AND G. WEISS 29
Cyclic composition operators on //2
P. S. BOURDON AND J. H. SHAPIRO 43
Representation theory of 17 (00)
ROBERT P. BOYER 55
Trace class integral kernels
CHRISTOPHER M. BRISLAWN 61
Products of operators with singular continuous spectra
GEON HO CHOE 65
Vll l CONTENTS
Joint hyponormality: A bridge between hyponormality and subnor-mality RAUL E. CURTO 69
Algebras generated by classical pseudodifferential operators on open Riemannian manifolds ALEXANDER DYNIN 93
A similarity invariant
C. K. FONG, E. A. NORDGREN, H. RADJAVI, AND P. ROSENTHAL 99
Two themes in index theory on singular varieties J. S. Fox, P. HASKELL, AND W. PARDON 103
CP-Duality for C*- and ^'-algebras ICHIRO FUJIMOTO 117
A note on special CSL algebras with nontrivial cohomology FRANK GILFEATHER 121
Actions of compact groups on C*-algebras E. C. GOOTMAN AND A. J. LAZAR 127
Parallel sums of operators W. L. GREEN AND T. D. MORLEY 129
The zero-two law and the peripheral spectrum of positive contractions in Banach lattice algebras J. J. GROBLER 135
Finite weight projections in von Neumann factors H. HALPERN, V. KAFTAL, AND L. ZSIDO 143
Toeplitz operators on noncommutative tori and their real valued index RONGHUI Ji 153
Close C*-algebras are A^-equivalent MAHMOOD KHOSHKAM 159
On derivations of operator algebras implemented by bounded operators EDWARD V. KISSIN 163
Composition operators and weighted polynomial approximation T. L. KRIETE, III AND B. D. MACCLUER 175
Seminormality for measure-theoretic composition operators ALAN LAMBERT 18 3
Local derivations and local automorphisms of 3&{X) D. R. LARSON AND A. R. SOUROUR 187
How to use primeness to describe properties of elementary operators MARTIN MATHIEU 19 5
Shifts on Krein spaces BRIAN W. MCENNIS 201
CONTENTS ix
Inner completely positive maps on von Neumann algebras JAMES A. MINGO 213
Isometries of nest algebras R. L. MOORE AND T. T. TRENT 219
Chern character for proper T-manifolds HITOSHI MORIYOSHI 221
Selfadjointness of quantum Hamiltonians of relativistic and nonrela-tivistic particles in electromagnetic fields MICHIRO NAGASE AND TOMIO UMEDA 235
Actions of compact groups on C*-algebras COSTEL PELIGRAD 243
Invariants for triangular AF algebras J. R. PETERS AND Y. T. POON 247
A^-theory and noncommutative homotopy theory N. CHRISTOPHER PHILLIPS 255
Equivalence bimodules in the representation theory of reductive groups R. J. PLYMEN 267
Spectral theory and sheaf theory IV MIHAI PUTINAR 273
On factorization of selfadjoint operator polynomials LEIBA RODMAN 295
The Kasparov groups for commutative C*-algebras and Spanier-White-head duality CLAUDE SCHOCHET 307
How are the Toeplitz C* -algebras of Reinhardt domains affected by taking pseudoconvex hulls? ALBERT J.-L. SHEU 323
Factor state extensions on O2 JOHN S. SPIELBERG 333
On the Dauns-Hofmann theorem SILVIU TELEMAN 337
Subnormal operators with a common invariant subspace T. T. TRENT AND W. WOGEN 343
C*-algebras and the classification of finite groups MARTIN E. WALTER 345
Bivariant Chern character I XIAOLU WANG 355
Composition operators acting on spaces of holomorphic functions on domains in Cn
WARREN R. WOGEN 361
X CONTENTS
Rigidity theorem and Buerling theorem KEREN YAN 367
On certain unitary operators and composition operators KEHEZHU 371
Preface
During the last twenty years operator theory has come of age. The subject has developed in several directions, using new and powerful methods that have led to the solution of basic problems thought to be inaccessible in the sixties. Some of these developments have made mutually enriching contact with other areas of mathematics, including algebraic topology and index theory, complex analysis in one and several variables, and probability theory.
This period has seen the full characterization of quasitriangular operators in terms of the Fredholm index, the classification of families of essentially normal operators via C*-algebraic extensions, and the consequences of the latter subject culminating in the unification of C*-algebraic ^-homology and A^-cohomology in the Kasparov A^-bifunctor. The invariant subspace problem has been solved for subnormal operators and related classes. The classical Weyl-von Neumann theorem has been vastly generalized to separable C*-algebras. The Ringrose problem on the multiplicity of nests of subspaces has been solved, using algebraic methods that have provided striking insight into the behavior of operators under similarity transforms. The classical perturbation theory for Schrodinger operators has been transformed and simplified by the use of path integrals and the Feynman-Kac formula. A rich theory of completely positive and completely bounded maps of C*-algebras has emerged, and this has had significant implications for operator theory, including dilation theory, the characterization of operators having annular spectral sets, and the partial solution of Sz.-Nagy's problem. The C*-algebras of Toeplitz operators associated with a large class of domains in Cn are now clearly understood. Finally, the structure of broad classes of reflexive operator algebras has been penetrated and put into the context of "noncommutative" spectral synthesis.
These algebraic methods are diverse, and they touch upon a broad area of mathematics. In addition to the interrelations alluded to above, there have been direct applications to systems theory, complex variables, and statistical mechanics. Moreover, significant problems and motivations have arisen from the subject's traditional underpinnings for partial differential equations. While it would not be possible or perhaps even desirable to attempt to unify these results and methods, it seemed an appropriate time to
xi
Xll PREFACE
summarize progress and examine the common points of view that now run through the subject. Thus, we organized the 1988 AMS Summer Institute on Operator Theory/Operator Algebras and Applications. The present two volumes contain papers representing most of the invited talks, as well as many of the seminar talks that were given there. We think that the points made above are amply borne out in these Proceedings.
William B. Arveson Ronald G. Douglas