Transcript C - GEOCITIES.ws

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 j1 )  [ρ(a ij )]i,n j1
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
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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