Operator algebraic characterization of the noncommutative Poisson boundary

Cyril Houdayer

doi:10.5802/crmath.715

Research article - Functional analysis

Operator algebraic characterization of the noncommutative Poisson boundary

Cyril Houdayer ¹

¹ École Normale Supérieure, Département de Mathématiques et Applications, Université Paris-Saclay, 45 rue d’Ulm, 75230 Paris Cedex 05, France

Comptes Rendus. Mathématique, Volume 363 (2025), pp. 199-204.

Abstract (Original language)
Résumé

We obtain an operator algebraic characterization of the noncommutative Furstenberg–Poisson boundary $\mathrm{L}(\Gamma ) \subset \mathrm{L}(\Gamma \curvearrowright B)$ associated with an admissible probability measure $\mu \in \mathrm{Prob}(\Gamma )$ 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 [14]. We apply this result in combination with previous works to provide further evidence towards Connes’ rigidity conjecture for higher rank lattices.

Nous obtenons une caractérisation en algèbres opérateurs de la frontière de Furstenberg–Poisson noncommutative $\mathrm{L}(\Gamma ) \subset \mathrm{L}(\Gamma \curvearrowright B)$ associée à une mesure de probabilité admissible $\mu \in \mathrm{Prob}(\Gamma )$ pour laquelle la $(\Gamma , \mu )$-frontière de Furstenberg–Poisson $(B, \nu _B)$ est uniquement $\mu $-stationnaire. Il s’agit d’une généralisation noncommutative du théorème de structure de Nevo–Sageev [14]. Nous appliquons ce résultat en combinaison avec des travaux antérieurs pour fournir des pistes supplémentaires afin de résoudre la conjecture de rigidité de Connes pour les réseaux de rang supérieur.

Received: 2024-11-04
Accepted: 2024-12-19
Published online: 2025-03-25

DOI: 10.5802/crmath.715

Classification: 20G25, 22D25, 46L10, 46L55
Keywords: Connes’ rigidity conjecture, higher rank lattices, noncommutative Furstenberg–Poisson boundaries, von Neumann algebras
Mots-clés : Conjecture de rigidité de Connes, réseaux de rang supérieur, frontières de Furstenberg–Poisson noncommutatives, algèbres de von Neumann

Author's affiliations:

Cyril Houdayer ¹

¹ École Normale Supérieure, Département de Mathématiques et Applications, Université Paris-Saclay, 45 rue d’Ulm, 75230 Paris Cedex 05, France

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Cyril Houdayer},
     title = {Operator algebraic characterization of the noncommutative {Poisson} boundary},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {199--204},
     publisher = {Acad\'emie des sciences, Paris},
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     year = {2025},
     doi = {10.5802/crmath.715},
     language = {en},
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Cyril Houdayer. Operator algebraic characterization of the noncommutative Poisson boundary. Comptes Rendus. Mathématique, Volume 363 (2025), pp. 199-204. doi : 10.5802/crmath.715. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.715/

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