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 con tinuous quantum dynamical semigroup T-t on a von Neumann algebra A with res pect to the Fock filtration is developed with the aid of a coordinate-free quantum stochastic calculus. Starting with the structure of the generator o f T-t, existence of canonical structure maps (in the sense of Evans and Hud son) is deduced and a quantum stochastic dilation of T-t is obtained throug h solving a canonical flow equation for maps on the right Fock module A X G amma(L-2(R+, k(0))), where k(0) is some Hilbert space arising from a repres entation of A'. This gives rise to a *-homomorphism j(t) of A. 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.