Hilbert Modules and Stochastic Dilation of a Quantum Dynamical Semigroup on a von Neumann Algebra

A general theory for constructing a weak Markov dilation of a uniformly continuous quantum dynamical semigroup T _t on a von Neumann algebra ? 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 ?⊗Γ(L ²(ℝ₊,k ₀)), where k ₀ is some Hilbert space arising from a representation of ?^′. This gives rise to a *-homomorphism j _t of ?. 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. Received: 15 June 1998/ Accepted: 4 March 1999