2002/02/19 by Г. Г. Амосов, Grigori G. Amosov, Amosov, Grigori G.
Mathematics · Physics and Astronomy · #46L #47D #81Q #Advanced Operator Algebra Research #FOS: Mathematics #Functional Analysis (math.FA) #Probability (math.PR) #Quantum Mechanics and Applications #advanced mathematical theories #math.FA #math.PR #msc:46L #msc:47D #msc:81Q
paper · pdf · doi:10.48550/arxiv.math/0202192
13 pages, the lecture given on the Conference on Quantum Probability and Infinite Dimensional Analysis, Cottbus, March 15-20, 2001
arxiv created 2002/02/19 · openalex publication_date 2002/02/19 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Stationary quantum stochastic process j is introduced as a *-homomorphism embedding an involutive graded algebra K=⊕i=1∞Ki into a ring of (abelian) cohomologies of the one-parameter group α consisting of *-automorphisms of certain operator algebra in a Hilbert space such that every x from Ki is translated into an additive i-α-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 E0-semigroup β on the von Neumann algebra \cal N associated with the markovian perturbation of K-flow j posseses the restriction β|_\cal N0, \cal N0⊂ \cal N, which is conjugate to the flow of Powers shifts β associated with j. It yields for β 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 L2(\mathbb R) all markovian cocycles of the group of shifts are described up to unitary equivalence of perturbations.