Quantum Markov Semigroups (Product Systems and Subordination)

TL;DR

This paper explores the structure of quantum Markov semigroups through product systems, establishing their Borel structures and analyzing subordination relations among semigroups.

Contribution

It introduces a natural Borel structure on product systems from quantum Markov semigroups and studies the dual systems and subordination order intervals.

Findings

Product systems from quantum Markov semigroups have a natural Borel structure.

The dual of a Borel product system also has a natural Borel structure.

The analysis characterizes the order interval of subordinate quantum Markov semigroups.

Abstract

We show that if a product system comes from a quantum Markov semigroup, then it carries a natural Borel structure with respect to which the semigroup may be realized in terms of a measurable representation. We show, too, that the dual product system of a Borel product system also carries a natural Borel structure. We apply our analysis to study the order interval consisting of all quantum Markov semigroups that are subordinate to a given one.