Peer-Reviewed Journal Details
Mandatory Fields
Lindsay, J. Martin; Wills, Stephen J.
2014
May
International Mathematics Research Notices
Quantum stochastic cocycles and completely bounded semigroups on operator spaces
Published
WOS: 3 ()
Optional Fields
Quantum stochastic cocycle CCR flow E-semigroup operator space operator system matrix space Schur action complete positivity
2014
11
3096
3139
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. Proof of the two key characterisations, 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.
Oxford
1687-0247
10.1093/imrn/rnt001
Grant Details