1. This research was supported by grant CNCSIS- code A 1065/2006 DILATIONS ON HILBERT C*- MODULES FOR C*- DYNAMICAL SYSTEMS MARIA JOIŢA, University of Bucharest TANIA – LUMINIŢA COSTACHE, University Politehnica of Bucharest MARIANA ZAMFIR, Technical University of Civil Engineering of Bucharest Bucharest

2. Mathematics in Engineering and Numerical Physics, BUCHAREST, Oct. 2006 KEYWORDS

3. Mathematics in Engineering and Numerical Physics, BUCHAREST, Oct. 2006 Definitions Definition 1 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 the topology determined by the norm ||·|| given by ||ξ|| = (  ξ, ξ  ) 1/2. If E and F are two Hilbert A -modules, we define L A (E, F) to be the set of all bounded module homomorphisms T : E → F for which there is a bounded module homomorphism T * : F → E such that  Tξ, η  =  ξ, T * η , for all ξ  E and η  F. We write L A (E) for C* -algebra L A (E, E).

4. Mathematics in Engineering and Numerical Physics, BUCHAREST, Oct. 2006 Definitions Definition 2 Let A be a C* -algebra and let E be a Hilbert C* -module. Denote by M n (A) the  -algebra of all n  n matrices over A. A completely positive linear map from A to L B (E) is a linear map ρ: A → L B (E) such that the linear map ρ (n) : M n (A) → M n (L B (E) ) defined by is positive for any integer positive n. We say that ρ is strict if (ρ(e λ )) λ is strictly Cauchy in L B (E), for some approximate unit (e λ ) λ of A. Definition 3 Let A be a C* -algebra and let α: A → A be an injective C* -morphism. A strict transfer operator for α is a strict completely positive linear map τ: A → A such that τ(α(a)) = a, for all a  A.

5. Mathematics in Engineering and Numerical Physics, BUCHAREST, Oct. 2006 The extension of a representation adapted to a strict transfer operator Proposition Let A be a C* -algebra, let φ : A → L B (E) be a nondegenerate representation of A on the Hilbert C* -module E over a C* -algebra B and let α : A → A be an injective C* -morphism which has a strict transfer operator τ. 1. There is a Hilbert B -module E τ, a representation Φ τ of A on E τ and an element V τ  L B (E, E τ ) such that: a) φ(a) = V τ * Φ τ (α(a))V τ, for all a  A; b) φ(τ(a)) = V τ * Φ τ (a)V τ, for all a  A; c) Φ τ (A)V τ E is dense in E τ. 2. If Φ is a representation of A on a Hilbert B -module F and V  L B (E, F) such that: a) φ(a) = V * Φ(α(a))V, for all a  A; b) φ(τ(a)) = V * Φ(a)V, for all a  A; c) Φ(A)VE is dense in F then there is a unitary operator U : E τ → F such that: UΦ τ (a) = Φ(a)U, for all a  A and UV τ = V.

6. Mathematics in Engineering and Numerical Physics, BUCHAREST, Oct. 2006 Definitions Let A be a C* -algebra and let α: A → A be an injective C* -morphism. Definition 3 A contractive (resp. isometric, resp. coisometric, resp. unitary) covariant representation of the pair (A, α) on a Hilbert C* -module is a triple (φ, T, E) consisting of a representation φ of A on a Hilbert C* -module E and a contractive (resp. isometric, resp. coisometric, resp. unitary) operator T in L B (E) such that T(φ(α(a)) = φ(a)T, for all a  A. Definition 4 Let (φ, T, E) be a contractive covariant representation of (A, α). A coisometric (resp. isometric, resp. unitary) covariant representation (Φ, V, F) of (A, α) on a Hilbert B -module F containing E as a complemented submodule is called dilation adapted to τ of (φ, T, E) if: E is invariant under Φ(A) and Φ(a)|E = φ(a), for all a  A, while P E V n |E = T n, for all n  0, where P E is the projector of F onto E.

7. Mathematics in Engineering and Numerical Physics, BUCHAREST, Oct. 2006 The main results Theorem 1 Let A be a C* -algebra, let α: A → A be an injective C* -morphism which has a strict transfer operator τ and let (φ, T, E) be a nondegenerate contractive covariant representation of (A, α) on a Hilbert C*-module E over a C* -algebra B. Then (φ, T, E) has a coisometric dilation adapted to τ, (Φ, V, F).

8. Mathematics in Engineering and Numerical Physics, BUCHAREST, Oct. 2006 The main results Theorem 2 Let A be a C* -algebra, let α: A → A be an injective C* -morphism and let (φ, T, E) be a contractive covariant representation of (A, α) on a Hilbert C* -module E over a C* -algebra B. Then (φ, T, E) has an isometric dilation (Φ,V, F). Further, if T is coisometric, then V is coisometric.

9. Mathematics in Engineering and Numerical Physics, BUCHAREST, Oct. 2006 The main results Corollary Let A be a C* -algebra, let α: A → A be an injective C* -morphism which has a strict transfer operator τ and let (φ, T, E) be a nondegenerate contractive covariant representation of (A, α) on a Hilbert C* -module E over a C* -algebra B. Then (φ, T, E) has a unitary dilation adapted to τ.

10. Mathematics in Engineering and Numerical Physics, BUCHAREST, Oct. 2006 ReferencesReferences  E. C. Lance, Hilbert C * -module. A toolkit for operator algebraists, London Mathematical Society Lecture Note Series 210, 1995;  P. S. Muhly, B. Solel, Extensions and Dilations for C*- dynamical Systems, arXiv: math. OA/0509506 v1, 22 Sept. 2005;  P. S. Muhly, B. Solel, Quantum Markov Processes (Correspondences and Dilations), International Journals of Mathematics, Vol. 13, No. 8, 2002;  P. S. Muhly, B. Solel, Tensor Algebras over C*- Correspondences: Representations, Dilations and C*- Envelopes, Journal of Functional Analysis 158, 1998;  B. Sz - Nagy, C. Foiaş, Harmonic Analysis of Operators in Hilbert Space, North-Holland, Amsterdam, 1970.