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Infinite Dimensional Analysis Quantum Probability and Related Topics

DILATION OF ARBITRARY SYMMETRIC QUANTUM DYNAMICAL SEMIGROUPS ON $B({\mathcal H})$

BISWARUP DAS

Published Aug 23, 2012• 1 citation
Publisher

Abstract

We prove the existence of Hudson–Parthasarathy dilation of a quantum dynamical semigroup on [Formula: see text], which is symmetric with respect to the canonical normal trace on it.
DOI
10.1142/s0219025712500178

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DILATION OF ARBITRARY SYMMETRIC QUANTUM DYNAMICAL SEMIGROUPS ON B(H)B({\mathcal H})B(H)
Hudson-Parthasarathy (H-P) type quantum stochastic dilation of a class of C 0 semigroups of completely positive maps ( quantum dynamical or Markov semigroups) on a von Neumann or C * algebra, with unbounded generators, is constructed under some assumptions on the semigroup and its generator.The assumption of symmetry with respect to a semifinite trace allows the use of Hilbert space techniques,while that of covariance with respect to an action of a Lie group on the algebra gives a better control on the domain of the generator.A dilation of the dynamical semigroup is obtained, under some further assumptions on the domain of the generator, with the help of a conjugation by a unitary quantum stochastic process satisfying Hudson-Parthasarathy equation in Fock space.
This is a continuation of the study of the theory of quantum stochastic dilation of completely positive semigroups on a von Neumann or C∗C^*C∗ algebra, here with unbounded generators. The additional assumption of symmetry with respect to a semifinite trace allows the use of the Hilbert space techniques, while the covariance gives rise to better handle on domains. An Evans-Hudson flow is obtained, dilating the given semigroup.