Operator algebraic characterization of the noncommutative Poisson boundary
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866908451046686720 |
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| author | Houdayer, Cyril |
| author_facet | Houdayer, Cyril |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_11707 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Operator algebraic characterization of the noncommutative Poisson boundary Houdayer, Cyril Operator Algebras Dynamical Systems Group Theory 20G25, 22D25, 46L10, 46L55 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. |
| title | Operator algebraic characterization of the noncommutative Poisson boundary |
| topic | Operator Algebras Dynamical Systems Group Theory 20G25, 22D25, 46L10, 46L55 |
| url | https://arxiv.org/abs/2410.11707 |