Hilbert modules and stochastic dilation of a quantum dynamical semigroup on a von Neumann algebra

Abstract

A general theory for constructing a weak Markov dilation of a uniformly continuous quantum dynamical semigroup T _t on a von Neumann algebra Α with respect to the Fock filtration is developed with the aid of a coordinate-free quantum stochastic calculus. Starting with the structure of the generator of T _t , existence of canonical structure maps (in the sense of Evans and Hudson) is deduced and a quantum stochastic dilation of T _t is obtained through solving a canonical flow equation for maps on the right Fock module Α ⊗ Γ(L²(R₊,k ₀)), where k₀ is some Hilbert space arising from a representation of Α'. This gives rise to a ∗-homomorphism j_t of Α . Moreover, it is shown that every such flow is implemented by a partial isometry-valued process. This leads to a natural construction of a weak Markov process (in the sense of [B-P]) with respect to Fock filtration.