Operator algebraic characterization of the noncommutative Poisson boundary

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.