MARIA JOI Ţ A, University of Bucharest

1. MARIA JOIŢA, University of Bucharest TANIA – LUMINIŢA COSTACHE, University Politehnica of Bucharest MARIANA ZAMFIR, Technical University of Civil Engineering of Bucharest COVARIANT REPRESENTATIONS ASSOCIATED WITH COVARIANT COMPLETELY n -POSITIVE LINEAR MAPS BETWEEN C* - ALGEBRAS This research was supported by grant CNCSIS-code A 1065/2006

2. 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: • ξ , η* =  η , ξ, for every ξ , η E; • ξ , ξ  0, for every ξ E; • 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 the topology 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. III Workshop on Coverings, Selections and Games in Topollogy, SERBIA, April 2007

3. 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 A which 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]] III Workshop on Coverings, Selections and Games in Topollogy, SERBIA, April 2007

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

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

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

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

8. Completely positive linear u –covariant map and non-degenerate covariant representation of a C* -dynamical system 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 ρbe a u –covariant non-degenerate completely positive linear map from A to LB(E). • Then there is a covariant representation (Φρ, vρ, Eρ) of (G, A, α) and Vρ in LB(E, Eρ) such that: • ρ(a) = Vρ*Φρ(a)Vρ, for all a  A; • {Φρ(a)Vρξ ; a  A, ξ E} spans a dense submodule of Eρ; • vρgVρ = Vρug , for all g  G. III Workshop on Coverings, Selections and Games in Topollogy, SERBIA, April 2007

9. Completely positive linear u –covariant map and non-degenerate covariant representation of a C* -dynamical system • 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: • ρ(a) = W*Φ(a)W, for all a  A; • {Φ(a)Wξ ; a  A, ξ E} spans a dense submodule of F; • vgW = Wug , for all g  G, then there is a unitary operator U in LB(Eρ, F) such that: • UΦρ(a) = Φ(a)U, for all a  A; • vgU = Uvρg , for all g  G; • W = UVρ. [Th.4.3, [2, Joiţa, case n =1]] III Workshop on Coverings, Selections and Games in Topollogy, SERBIA, April 2007

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

11. 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). Theorem1 The map T  ρT from [0, I]ρ to [0, ρ] is an affine order isomorphism. III Workshop on Coverings, Selections and Games in Topollogy, SERBIA, April 2007

12. The main results Theorem2 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). • Then there is (Φρ, vρ, Eρ) a covariant representation of (G, A, α) on a Hilbert B -moduleEρ, anisometry Vρ: E → Eρ and such that: • , 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; • vρgVρ = Vρug, for all g  G. III Workshop on Coverings, Selections and Games in Topollogy, SERBIA, April 2007

13. The main results • 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: • , for all a  A and for all i, j = 1, 2, …, n, is a positive element in Mn(LB(E)) and ; • {ψ(a)Wξ ; a  A, ξE} is dense in F; • , for all a  A and i, j = 1, 2, …, n; • wgW = Wug, for g  G then there is a unitary operator U:Eρ → F such that: • ψ(a) = UΦρ(a)U*, for all a  A; • W = UVρ; • Sij = UTρijU*, for all i, j =1, 2, …, n; • wg = UvρgU*, for all g  G. III Workshop on Coverings, Selections and Games in Topollogy, SERBIA, April 2007

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

15. 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: • for all f A α G and for all i, j =1, 2, …, n, is a positive element in Mn(LB(E)) and • {Φφ(f)Vφξ ; f  A αG, ξE} is dense in Eφ; • for all f A α G and for all i, j =1, 2, …, n. III Workshop on Coverings, Selections and Games in Topollogy, SERBIA, April 2007

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

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

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