Poisson boundaries of II$_1$ factors

Abstract

We introduce Poisson boundaries of II1 factors with respect to density operators that give the traces. The Poisson boundary is a von Neumann algebra that contains the II1 factor and is a particular example of the boundary of a unital completely positive map as introduced by Izumi. Studying the inclusion of the II1 factor into its boundary we develop a number of notions, such as double ergodicity and entropy, that can be seen as natural analogues of results regarding the Poisson boundaries introduced by Furstenberg. We use the techniques developed to answer a problem of Popa by showing that all finite factors satisfy the MV-property. We also extend a result of Nevo by showing that property (T) factors give rise to an entropy gap.

Properly Proximal von Neumann Algebras

“…The advantage of viewing M ⊗ Con N op as a subspace of the dual of K(L 2 N, L 2 M ) is that we may then use techniques for linear functionals, e.g., Jordan decomposition, when working with elements in M ⊗ Con N op , see, e.g., Lemma 5.4 below or [DP20] where similar techniques are developed.…”