Vol. 137, No. 2, 1989

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

Alain Connes and E. J. Woods

Vol. 137 (1989), No. 2, 225–243

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).

Mathematical Subject Classification 2000
Primary: 46L50, 46L50
Secondary: 28D15, 46L35, 60J15, 60J50
Received: 2 May 1988
Published: 1 April 1989
Alain Connes
E. J. Woods