Quantum stochastic cocycles and completely bounded semigroups on operator spaces

Lindsay, J. Martin; Wills, Stephen J.

Quantum stochastic cocycles and completely bounded semigroups on operator spaces

Files

2011iii21sjwRevQS_CBonOS_I.pdf(346.04 KB)

Accepted Version

Date

2013-03-06

Authors

Lindsay, J. Martin

Wills, Stephen J.

Publisher

Oxford University Press

Published Version

https://doi.org/10.1093/imrn/rnt001

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 analog 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. Proof of the two key characterizations, namely that of completely positive contraction cocycles on a C*-algebra, and contraction cocycles on a Hilbert space, involves all of the analysis undertaken here. 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.

Keywords

Operator space analysis , Quantum stochastic cocycles

Citation

Lindsay, J. M. and Wills, S.J. (2013) 'Quantum stochastic cocycles and completely bounded semigroups on operator spaces', International Mathematics Research Notices, 2014(11), pp. 3096-3139. https://doi.org/10.1093/imrn/rnt001