Hyperfinite von Neumann algebras and poisson boundaries of time dependent random walks

Abstract

We consider the problem of characterizing Poisson boundaries of group-invariant time-dependent Markov random walks on locally compact groups G. We show that such Poisson boundaries, which by construction are naturally G-spaces, are amenable and approximately transitive (see Definition 1.1 and Theorem 2.2). We also establish a relationship between von Neumann algebras and Poisson boundaries when G = R or Z. More precisely, there is naturally associated to an eigenvalue list for an ITPFI factor M, a group-invariant time-dependent Markov random walk on R whose Poisson boundary is the flow of weights for M (Theorem 3.1). © 1989 by Pacific Journal of Mathematics.