Structure of block quantum dynamical semigroups and their product systems

Article Type

Research Article

Publication Title

Infinite Dimensional Analysis, Quantum Probability and Related Topics

Abstract

Paschke's version of Stinespring's theorem associates a Hilbert C∗-module along with a generating vector to every completely positive map. Building on this, to every quantum dynamical semigroup (QDS) on a C∗-algebra one may associate an inclusion system E = (Et) of Hilbert-modules with a generating unit ζ = (ζt). Suppose B is a von Neumann algebra, consider M2(B), the von Neumann algebra of 2 × 2 matrices with entries from B. Suppose (φt)t≥0 with φt = {equation presented}, is a QDS on M2(B) which acts block-wise and let (Eti) t≥0 be the inclusion system associated to the diagonal QDS (φti) t≥0 with the generating unit (ζti) t≥0,i = 1, 2. It is shown that there is a contractive (bilinear) morphism T = (Tt)t≥0 from (Et2) t≥0 to (Et1) t≥0 such that ψt(a) = (ζt1,Ttaζt2) for all a ∈ B. We also prove that any contractive morphism between inclusion systems of von Neumann B-B modules can be lifted as a morphism between the product systems generated by them. We observe that the E0-dilation of a block quantum Markov semigroup (QMS) on a unital C∗-algebra is again a semigroup of block maps.

DOI

10.1142/S0219025720500010

Publication Date

3-1-2020

Comments

Open Access, Green

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