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arXiv · 2410.11707

Operator algebraic characterization of the noncommutative Poisson boundary

Cyril Houdayer
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Abstract

We obtain an operator algebraic characterization of the noncommutative Furstenberg-Poisson boundary $\operatorname{L}(Γ) \subset \operatorname{L}(Γ\curvearrowright B)$ associated with an admissible probability measure $μ\in \operatorname{Prob}(Γ)$ for which the $(Γ, μ)$-Furstenberg-Poisson boundary $(B, ν_B)$ is uniquely $μ$-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.

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Cyril Houdayer. 2025-07-15. Operator algebraic characterization of the noncommutative Poisson boundary. https://doi.org/10.5802/crmath.715

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