Lindsay, J. Martin and Wills, Stephen J. (2014) *Quantum stochastic cocycles and completely bounded semigroups on operator spaces.* International Mathematics Research Notices, 2014 (11). pp. 3096-3139. ISSN 1073-7928

## 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.

Item Type: | Article |

Journal or Publication Title: | International Mathematics Research Notices |

Uncontrolled Keywords: | Quantum stochastic cocycle, CCR flow, E-semigroup,operator space, operator system, matrix space, completely bounded, completely positive. ; CCR flow ; E-semigroup ; operator space ; operator system ; matrix space ; completely bounded ; completely positive |

Subjects: | |

Departments: | Faculty of Science and Technology > Mathematics and Statistics |

ID Code: | 58420 |

Deposited By: | ep_importer_pure |

Deposited On: | 18 Sep 2012 12:09 |

Refereed?: | Yes |

Published?: | Published |

Last Modified: | 07 Jan 2015 17:54 |

Identification Number: | |

URI: | http://eprints.lancs.ac.uk/id/eprint/58420 |
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