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

摘要

A general theory for constructing a weak Markov dilation of a uniformly continuous quantum dynamical semigroup T_t on a von Neumann algebra A 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 A (direct X) #GAMMA# (L~2(R_+, k_0)), where k_0 is some Hilbert space arising from a representation 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.