Poisson boundaries of finite von Neumann algebras

Abstract

Poisson boundaries of groups plays a major role in the study of group actions on measure spaces. In this work, we study noncommmutative Poisson boundaries of finite von Neumannalgebras. We prove a noncommutative analogue of the double ergodicity theorem due to V.Kaimanovich and give applications to the study of derivations on a finite von Neumann algebra,and the similarity problem. We also prove a boundary rigidity theorem, using double ergodicity. We also define and study the notions of noncommutative Avez entropy, and noncommutative Fustenberg entropy, and show an entropy gap theorem for von Neumann algebras with property(T).