Quantum stochastic cocycles and completely bounded semigroups on operator spaces

TL;DR

This paper analyzes quantum stochastic cocycles within operator spaces, establishing correspondences with semigroups and providing a broader perspective that overcomes previous technical limitations.

Contribution

It introduces a new operator space framework for quantum stochastic cocycles, linking them to semigroups and bypassing domain restrictions of earlier differential equation approaches.

Findings

Established one-to-one correspondences between cocycles and semigroups.

Extended analysis to various classes of cocycles including completely positive and contractive.

Provided a broader operator space perspective for quantum stochastic calculus.

Abstract

An operator space analysis of quantum stochastic cocycles is undertaken. These are cocycles with respect to an ampliated CCR flow, adapted to the associated filtration of subspaces, or subalgebras. They form a noncommutative analogue of stochastic semigroups in the sense of Skorohod. One-to-one correspondences are established between classes of cocycle of interest and corresponding classes of one-parameter semigroups on associated matrix spaces. Each of these 'global' semigroups may be viewed as the expectation semigroup of an associated quantum stochastic cocycle on the corresponding matrix space. The classes of cocycle covered include completely positive contraction cocycles on an operator system, or C*-algebra; completely contractive cocycles on an operator space; and contraction operator cocycles on a Hilbert space. As indicated by Accardi and Kozyrev, the Schur-action matrix semigroup viewpoint circumvents technical (domain) limitations inherent in the theory of quantum stochastic differential equations. An infinitesimal analysis of quantum stochastic cocycles from the present wider perspective is given in a sister paper.