×

Quantum stochastic cocycles and completely bounded semigroups on operator spaces. (English) Zbl 1408.46054

Summary: 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 \(C^\ast\)-algebra, and contraction cocycles on a Hilbert space, involves all of the analysis undertaken here. As indicated by L. Accardi and S. V. Kozyrev [Chaos Solitons Fractals 12, No.14–15, 2639–2655 (2001; Zbl 1016.46041)], the Schur-action matrix semigroup viewpoint circumvents technical (domain) limitations inherent in the theory of quantum stochastic differential equations.

MSC:

46L07 Operator spaces and completely bounded maps
46L57 Derivations, dissipations and positive semigroups in \(C^*\)-algebras
81S25 Quantum stochastic calculus
47L25 Operator spaces (= matricially normed spaces)

Citations:

Zbl 1016.46041
PDFBibTeX XMLCite
Full Text: DOI arXiv Link