Stationary quantum stochastic processes from the cohomological point of view

Abstract

Stationary quantum stochastic process j is introduced as a *-homomorphism embedding an involutive graded algebra $\tilde K=\oplus_{i=1}^{\infty}K_i$ into a ring of (abelian) cohomologies of the one-parameter group $\alpha$ consisting of *-automorphisms of certain operator algebra in a Hilbert space such that every x from $K_i$ is translated into an additive $i-\alpha$-cocycle j(x). It is shown that (noncommutative) multiplicative markovian cocycle defines a perturbation of the stationary quantum stochastic process in the sense of such definition. The $E_0$-semigroup $\tilde \beta$ on the von Neumann algebra $\cal N$ associated with the markovian perturbation of K-flow j posseses the restriction $\tilde \beta |_{{\cal N}_0}, {\cal N}_0\subset {\cal N}$, which is conjugate to the flow of Powers shifts $\beta$ associated with j. It yields for $\tilde \beta$ an analogue of the Wold decomposition for classical stochastic process on completely nondeterministic and deterministic parts. The examples of quantum stationary stochastic processes on the algebras of canonical commutation, anticommutation and square of white noise relations are considered. In the model situation of the space $L^2(\mathbb R)$ all markovian cocycles of the group of shifts are described up to unitary equivalence of perturbations.