Operator algebraic characterization of the noncommutative Poisson boundary

TL;DR

This paper provides an operator algebraic characterization of the noncommutative Poisson boundary, extending classical results to a noncommutative setting and contributing to the understanding of rigidity phenomena in higher rank lattices.

Contribution

It offers a new operator algebraic framework for the noncommutative Poisson boundary, generalizing Nevo-Sageev's structure theorem to the noncommutative case.

Findings

Characterization of the noncommutative Furstenberg-Poisson boundary

Extension of classical structure theorems to noncommutative settings

Supporting evidence for Connes' rigidity conjecture in higher rank lattices

Abstract

We obtain an operator algebraic characterization of the noncommutative Furstenberg-Poisson boundary $\operatorname{L}(\Gamma) \subset \operatorname{L}(\Gamma \curvearrowright B)$ associated with an admissible probability measure $\mu \in \operatorname{Prob}(\Gamma)$ for which the $(\Gamma, \mu)$-Furstenberg-Poisson boundary $(B, \nu_B)$ is uniquely $\mu$-stationary. This is a noncommutative generalization of Nevo-Sageev's structure theorem [NS11]. We apply this result in combination with previous works to provide further evidence towards Connes' rigidity conjecture for higher rank lattices.