TY - JOUR

T1 - Quantum stochastic cocycles and completely bounded semigroups on operator spaces

AU - Lindsay, J. Martin

AU - Wills, Stephen J.

PY - 2014

Y1 - 2014

N2 - 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.

AB - 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.

KW - Quantum stochastic cocycle, CCR flow, E-semigroup,operator space, operator system, matrix space, completely bounded, completely positive.

KW - CCR flow

KW - E-semigroup

KW - operator space

KW - operator system

KW - matrix space

KW - completely bounded

KW - completely positive

U2 - 10.1093/imrn/rnt001

DO - 10.1093/imrn/rnt001

M3 - Journal article

VL - 2014

SP - 3096

EP - 3139

JO - International Mathematics Research Notices

JF - International Mathematics Research Notices

SN - 1073-7928

IS - 11

ER -