Operator algebraic characterization of the noncommutative Poisson boundary

Abstract

We obtain an operator algebraic characterization of the noncommutative Furstenberg-Poisson boundary $L(\hat{I}“)\subset L(\hat{I}“\curvearrowright B)$ associated with an admissible probability measure $\hat{I}¼\in \mathrm{Prob}(\hat{I}“)$ for which the $(\hat{I}“,\hat{I}¼)$-Furstenberg-Poisson boundary $(B,\hat{I}{½}_B)$ is uniquely $\hat{I}¼$-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.

6 pages