# 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

## Recommended Citation

Rajarama Bhat, B. V. and Vijaya Kumar, U., "Structure of block quantum dynamical semigroups and their product systems" (2020). *Journal Articles*. 359.

https://digitalcommons.isical.ac.in/journal-articles/359

## Comments

Open Access, Green