Dilation of a class of quantum dynamical semigroups

Abstract

Hudson-Parthasarathy (H-P) type quantum stochastic dilation of a class of C₀ 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.