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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 This research was supported by grant CNCSIS- code A 1065/2006 KEYWORDS 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 LA(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 LA(E) for C* -algebra LA(E, E). 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 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 ρ (n) ([a ij ]i,n j1 ) [ρ(a ij )]i,n j1 is positive for any integer positive 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 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. 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 → LB(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 Eτ, a representation Φτ of A on Eτ and an element Vτ LB(E, Eτ ) such that: a) φ(a) = Vτ*Φτ(α(a))Vτ , for all a A; b) φ(τ(a)) = Vτ*Φτ(a)Vτ , for all a A; c) Φτ(A)VτE is dense in Eτ. 2. If Φ is a representation of A on a Hilbert B -module F and V LB(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 : Eτ → F such that: UΦτ(a) = Φ(a)U, for all a A and UVτ = V. 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 LB(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 PEVn|E = Tn, for all n 0, where PE is the projector of F onto E. 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). 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. 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. Mathematics in Engineering and Numerical Physics, BUCHAREST, Oct. 2006 References 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. Mathematics in Engineering and Numerical Physics , BUCHAREST, Oct. 2006