Poisson boundaries of II$_1$ factors

TL;DR

This paper introduces Poisson boundaries for II$_1$ factors using density operators, developing new notions like ergodicity and entropy, and applies these to solve problems related to finite factors and property (T).

Contribution

It defines Poisson boundaries for II$_1$ factors and develops a framework for analyzing their properties, including ergodicity and entropy, with applications to longstanding problems.

Findings

All finite factors satisfy the MV-property.

Property (T) factors exhibit an entropy gap.

The framework generalizes Poisson boundary concepts to operator algebras.

Abstract

We introduce Poisson boundaries of II$_1$ factors with respect to density operators that give the traces. The Poisson boundary is a von Neumann algebra that contains the II$_1$ factor and is a particular example of the boundary of a unital completely positive map as introduced by Izumi. Studying the inclusion of the II$_1$ 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.