MARIA JOI Ţ A, University of Bucharest

This research was supported by grant CNCSIS-code A 1065/2006

COVARIANT REPRESENTATIONS ASSOCIATED WITH COVARIANT

COMPLETELY n -POSITIVE LINEAR MAPS BETWEEN C* - ALGEBRAS

MARIA JOIŢA, University of Bucharest

TANIA – LUMINIŢA COSTACHE, University Politehnica of Bucharest

MARIANA ZAMFIR, Technical University of Civil Engineering of

Bucharest

III Workshop on Coverings, Selections and Games in Topollogy, SERBIA, April 2007

C*-module Hilbert over C* -algebra Definition 1 Let A be a C* -algebra. A pre-Hilbert A -module is a complex vector space E which is also a right A -module, compatible with the complex algebra structure, equipped with an A -valued inner product ·, ·: E E → A which is C- and A –linear in its second variable and satisfies the following relations: 1. ξ , η* = η , ξ, for every ξ , η E; 2. ξ , ξ 0, for every ξ E; 3. ξ , ξ = 0 if and only if ξ = 0. We say that E is a Hilbert A -module if E is complete with respect to thetopology determined by the norm || · || given by || ξ || = (ξ , ξ )1/2. Notations If E and F are two Hilbert A -modules, we make the following notations: BA(E, F) is the Banach space of all bounded module homomorphisms from E to F LA(E, F) is the set of all maps T BA(E, F) for which there is a map T* BA(F, E)

such that Tξ , η = ξ , T*η, for all ξ E and for all η F.


C*-module Hilbert over C*-algebra

In general, BA(E, F) LA(E, F) and so the theory of Hilbert C* -modules and the theory of Hilbert spaces are different.

E# is the Banach space of all bounded module homomorphisms from E to Awhich becomes a right A -module, where the action of A on E# is defined by (aT)(ξ) = a*(Tξ), for a A, T E# , ξ E . We say that E is self-dual if E = E# as right A-modules.

If E and F are self-dual, then BA(E, F) = LA(E, F) [Prop. 3.4., Paschke, [8]]. Any bounded module homomorphism T from E to F extends uniquely to a

bounded homomorphism from E# to F# [Prop. 3.6., Paschke, [8]]. If A is a W* -algebra, E# becomes a self-dual Hilbert A –module [Th. 3.2., Paschke, [8]]

T~


The construction “Lin, Paschke, Tsui” of the self-dual Hilbert B**-module

Let E be a Hilbert B -module and let B** be the enveloping W* -algebra of B. On the algebraic tensor product E alg B** we define the action of right B**-module by (ξ b)c = ξ bc, for ξ E and b, c B**and the B** -valued

inner-product by . The quotient module

E alg B**/NE , where NE = {ζ E alg B**; [ζ, ζ] = 0}, becomes a pre-Hilbert

B**-module. The Hilbert C* -module obtained by the completion

of E alg B**/NE with respect to the norm induced by the inner product [·, ·] is

called the extension of E by the C* -algebra B**.

The self-dual Hilbert B** -module is denoted by and we

consider E as embedded in without making distinction.

#Ealg N|BE E~

E~

n

1i

m

1jjji i

m

1jjj

n

1ii i cη,ξbcη,bξ

E**

alg N|BE


The construction “Lin, Paschke, Tsui” of the self-dual Hilbert B** -module

Let E and F be two Hilbert modules over C* -algebra B. Then any operator T BB(E, F) extends uniquely to a bounded module

homomorphism from to such that || T || = || || [Prop. 3.6., Paschke, [8]].

If T LB(E, F), then .

A * -representation of a C* -algebra A on the Hilbert B -module E is a

-morphism Φ from A to LB(E) (meaning LB(E, E)). This representation induces a

representation of A on denoted by , for all a A .

The representation Φ is non-degenerate if Φ(A)E is dense in E.

E~

F~

T~

T~

Φ~

E~

Φ(a)(a)Φ~

T T

~


Completely positive and n –positive linear map and C* -dynamical system

Definition 2 Let A and B be two C* -algebras and E a Hilbert B-module. Denote by Mn(A) the * -algebra of all n n matrices over A. A completely positive linear map from A to LB(E) is a linear map ρ: A → LB(E) such that the linear map ρ(n): Mn(A) → Mn(LB(E)) defined by

is positive for any positive integer n. We say that ρ is strict if (ρ(eλ))λ is strictly Cauchy in LB(E), for some approximate unit (eλ)λ of A.

Definition 3 A completely n -positive linear map from A to LB(E) is a n n

matrix of linear maps from A to LB(E) such that the map

ρ: Mn(A) → Mn(LB(E)) defined by is completely positive

Definition 4 A triple (G, A, α) is a C* -dynamical system if G is a locally compact group, A is a C* -algebra and α is a continuous action of G on A.

n1ji,ij

n1ji,ij

(n) ])ρ(a[)]a([ρ

n1ji,ij]ρ[

n1ji,ijij

n1ji,ij )](a[ρ)]ρ([a


Completely positive linear u –covariant map and non-degenerate covariant representation of a

C* -dynamical system

Definition 5 Let (G, A, α) be a C* -dynamical system, let B be a C* -algebra and let u be a unitary representation of G on a Hilbert B –module E. A completely positive linear map ρ from A to LB(E) is u –covariant with respect to the C* -dynamical system (G, A, α) if ρ(αg(a)) = ug ρ(a) ug

*, for all a A and g G.

Definition 6 A covariant non-degenerate representation of a C* -dynamical system (G, A, α) on a Hilbert B –module E is a triple (Φ, v, E), where Φ is a non-degenerate continuous * -representation of A on E, v is a unitary representation of G on E and Φ(αg(a)) = vg Φ(a) vg

*, for all a A and g G.




Proposition Let (G, A, α) be a C* -dynamical system, let u be

a unitary representation of G on a Hilbert module E over a C* -algebra B, let ρ be a u –covariant non-degenerate completely positive linear

map from A to LB(E).

1. Then there is a covariant representation (Φρ, vρ, Eρ) of (G, A, α)

and Vρ in LB(E, Eρ) such that:

i. ρ(a) = Vρ*Φρ(a)Vρ , for all a A;

ii. {Φρ(a)Vρξ ; a A, ξ E} spans a dense submodule of Eρ;

iii. vρgVρ = Vρug , for all g G.




2. If F is a Hilbert B –module, (Φ, v, F) is a covariant representation of (G, A, α) and W is in LB(E, F) such that:

a) ρ(a) = W*Φ(a)W, for all a A;

b) {Φ(a)Wξ ; a A, ξ E} spans a dense submodule of F;

c) vgW = Wug , for all g G,

then there is a unitary operator U in LB(Eρ, F) such that:

i. UΦρ(a) = Φ(a)U, for all a A;

ii. vgU = Uvρg , for all g G;

iii. W = UVρ. [Th.4.3, [2, Joiţa, case n =1]]


The main results

Let A be a C* -algebra, let E be a C* -module Hilbert over a C* -algebra B and let ρ: A → LB(E) be a strict completely positive linear u -covariant map.

Notation C(ρ) the C* -subalgebra of generated by

[0, ρ] the set of all strict completely positive linear u –covariant maps θ

from A to LB(E) such that θ ρ (that is, ρ – θ is a strict completely positive

linear u –covariant map from A to LB(E)).

[0, I]ρ the set of all elements T in C(ρ) such that 0 T .

)E~

(L ρB **

A}a)((E),L|V~

(a)Φ~

TV~

T,v~v~T (a)T,Φ~

(a)Φ~

T);E~

(LT{ BEρρ*ρ

ρg

ρgρρρB **

ρE~I


The main results

Lemma Let T C(ρ) positive. Then the map ρT

defined by

is a strict completely positive linear u –covariant map

from A to LB(E).

Theorem 1 The map T ρT from [0, I]ρ to [0, ρ] is

an affine order isomorphism.

Eρρ*ρT |V

~(a)Φ

~TV

~(a)ρ


The main results Theorem 2 Let (G, A, α) be a C* -dynamical system, let u be a unitary representation of G on a Hilbert C* -module E over a C* -algebra B , let be a completely n-positive linear u -covariant map relative to the dynamical system (G, A, α) from A to LB(E).

1. Then there is (Φρ, vρ, Eρ) a covariant representation of (G, A, α) on a Hilbert

B -module Eρ, an isometry Vρ : E → Eρ and

such that: i. , for all a A and for all i, j = 1, 2, …, n,

is a positive element in Mn(LB(E)) and

{Φρ(a)Vρξ ; a A, ξ E} is dense in Eρ;

, for all a A and i, j = 1, 2, …, n; vρ

gVρ = Vρug, for all g G.

n1ji,ij]ρ[ρ

))(Gv~)(AΦ~

(M]T[ ρρn

n1ji,

ρij

(E)L|V~

(a)Φ~

TV~

BEρρρijρ

n1ji,Eρ

ρijρ ]|V

~TV

~[

) E

~ (L

n

1k

ρkk

ρ**B

nIT

Eρρρijρij |V

~(a)Φ

~T V

~(a)ρ


The main results2. If (ψ, w, F) is another covariant representation of (G, A, α) on a Hilbert

B-module F, W: E → F is an isometry and such that:

i. , for all a A and for all i, j = 1, 2, …, n,

is a positive element in Mn(LB(E)) and ;

ii. {ψ(a)Wξ ; a A, ξ E} is dense in F;

iii. , for all a A and i, j = 1, 2, …, n;

iv. wgW = Wug, for g G

then there is a unitary operator U: Eρ → F such that:

a) ψ(a) = UΦρ(a)U*, for all a A;

b) W = UVρ;

c) Sij = UTρijU*, for all i, j =1, 2, …, n;

d) wg = UvρgU*, for all g G.

))(Gw~)(Aψ~(M][S nn

1ji,ij

(E)L|W~

(a)ψ~SW~

BEij

n1ji,Eij ]|W

~SW

~[

)F~

(L

n

1kkk

**B

nIS

Eijij |W~

(a)ψ~S W~

(a)ρ


Crossed product associated to a C* -dynamical system

Definition 7 Let (Φ, v, E) be a covariant representation (possibly degenerate) of the dynamical system (G, A, α) on a Hilbert B –module E. Then

is a * -representation of Cc(G, A) on E called the integrated form of (Φ, v, E).

Definition 8 Let (G, A, α) be a dynamical system. For f Cc(G, A) we define the norm on Cc(G, A): || f || = sup {|| Φ v(f) ||; (Φ, v, E) is a covariant representation of (G, A, α)} called the universal norm. The completion of Cc(G, A) with respect to || · || is a C* -algebra called the crossed product of A by G and is denoted by A α G.

dgvΦ(f(g))v(f)Φ g

G


The main results

Let be a completely n –positive linear u –covariant non-degenerate map

with respect to a C* -dynamical system (G, A, α).By [Prop. 4.5., 2], there is a uniquely completely n –positive linear map

from A α G to LB(E) such that

for all f Cc(G, A) and for all i, j = 1, 2, …, n.

By [Th. 2.2, 5] there is a representation Φφ of A α G on Eφ, an isometry Vφ : E → Eφ

and such that:

a) for all f A α G and for all i, j =1, 2, …, n,

is a positive element in Mn(LB(E)) and

b) {Φφ(f)Vφξ ; f A α G, ξ E} is dense in Eφ;

c) for all f A α G and for all i, j =1, 2, …, n.

dgu(f(g))ρ(f) g

G

ijij

n1ji,ij]ρ[ρ

n1ji,ij][

))G(AΦ~

(M]T[ nn

1ji,ij

(E)L|V~

(f)Φ~

TV~

BEij

n1ji,Eij ]|V

~TV

~[

) E

~ (L

n

1kkk

**B

nIT

Eijij |V~

(f)Φ~

T V~

(f)ρ


The main results

By Theorem 2 there is (Φρ, vρ, Eρ) a covariant representation of (G, A, α) on a Hilbert B -module Eρ. By [Prop. 2.39, 10], (Φρ vρ, Eρ) is a representation of A α G on Eρ such that

for all f Cc(G, A), g G.

Proposition Let be a completely n –positive linear u –covariant

non-degenerate map and let be a uniquely completely n –positive linear map from A α G to LB(E) given by [Prop. 4.5., 2]. Then and are unitarily equivalent.

dgv(f(g))Φ(f)vΦ gρ

G

ρρ

ρ

n1ji,ij]ρ[ρ

n1ji,ij][

)][T,V,E,(Φ n1ji,ij

)][T,V,E,v(Φ n

1ji,ρ

ijρρρ

ρ


References

Arveson, W., Subalgebras of C* -algebras, Acta Math., 1969;

Joiţa, M., Completely multi-positive linear maps between locally C* -algebras and representations on Hilbert modules, Studia Math., 2006;

Joiţa, M., A Radon - Nikodym theorem for completely multi positive linear maps and its aplications, Proceedings of the 5–th International Conference on Topological Algebras and Applications, Athens, Greece, 2005 (to appear);

Joiţa, M., A Radon - Nikodym theorem for completely n-positive linear maps on pro -C* -algebras and its applications, Publicationes Mathematicae Debrecen;

Joiţa, M., Costache, T. L., Zamfir, M., Representations associated with completelly n-positive linear maps between C* -algebras, Stud. Cercet. Stiint., Ser. Mat., 2006;


References

Lance, E. C., Hilbert C* -module. A toolkit for operator algebraists, London Mathematical Society Lecture Note Series 210, 1995;

Lin, H., Bounded module maps and pure completely positive maps, J. Operator Theory, 1991;

Paschke, W. L., Inner product modules over B* -algebras, Trans. Amer. Math. Soc., 1973;

Tsui, S. K., Completely positive module maps and completely positive extreme maps, Proc. Amer. Math. Soc., 1996;

Williams, D., Crossed products of C* -algebras, Mathematical Surveys and monographs, 2006.

MARIA JOI Ţ A, University of Bucharest

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