Rigidity results for Bernoulli actions and their von Neumann algebras

Rigidity results for Bernoulli actions and their von Neumann algebras
[Rigidité pour les shifts de Bernoulli et leurs algèbres de von Neumann]

Vaes, Stefaan

Séminaire Bourbaki : volume 2005/2006, exposés 952-966, Astérisque, no. 311 (2007), Exposé no. 961, pp. 237-294.

Résumé
Abstract

Par des méthodes très originales d’algèbres d’opérateurs, Sorin Popa a démontré que si un groupe $G$ ayant la propriété (T) de Kazhdan agit par shift de Bernoulli, alors la partition en orbites se souvient entièrement du groupe et de l’action. Ces informations sont même essentiellement retenues par l’algèbre de von Neumann de ce système dynamique, ce qui constitue dans la littérature le premier résultat de rigidité forte pour les algèbres de von Neumann. Avec ces mêmes méthodes, Popa construit également des facteurs de type II $_{1}$ ayant un groupe fondamental dénombrable arbitraire.

Using very original methods from operator algebras, Sorin Popa has shown that the orbit structure of the Bernoulli action of a property (T) group, completely remembers the group and the action. This information is even essentially contained in the crossed product von Neumann algebra. This is the first von Neumann strong rigidity theorem in the literature. The same methods allow Popa to obtain II $_{1}$ factors with prescribed countable fundamental group.

EuDML MR Zbl | 3 citations dans Numdam

Classification : 46L35, 37A20, 46L10
Keywords: superrigidity, Bernoulli action, property (T), classification of von Neumann algebras, II$_1$ factor, fundamental group
Mot clés : superrigidité, shift de Bernoulli, propriété (T), classification d’algèbres de von Neumann, facteur II$_1$, groupe fondamental

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Vaes, Stefaan. Rigidity results for Bernoulli actions and their von Neumann algebras, dans Séminaire Bourbaki : volume 2005/2006, exposés 952-966, Astérisque, no. 311 (2007), Exposé no. 961, pp. 237-294. http://www.numdam.org/item/SB_2005-2006__48__237_0/

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