Operator algebraic characterization of the noncommutative Poisson boundary

We obtain an operator algebraic characterization of the noncommutative Furstenberg-Poisson boundary $\mathrm{L}(\mathrm{\Gamma })\subset \mathrm{L}(\mathrm{\Gamma }\curvearrowright B)$ associated with an admissible probability measure $\mu \in \mathrm{Prob}(\mathrm{\Gamma })$ for which the $(\mathrm{\Gamma },\mu )$-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.