Rigidity results for Bernoulli actions and their von Neumann algebras
Séminaire Bourbaki : volume 2005/2006, exposés 952-966, Astérisque no. 311  (2007), Talk no. 961, p. 237-294

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.

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.

Classification:  46L35,  37A20,  46L10
Keywords: superrigidity, Bernoulli action, property (T), classification of von Neumann algebras, II 1 factor, fundamental group
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     author = {Vaes, Stefaan},
     title = {Rigidity results for Bernoulli actions and their von Neumann algebras},
     booktitle = {S\'eminaire Bourbaki : volume 2005/2006, expos\'es 952-966},
     author = {Collectif},
     series = {Ast\'erisque},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {311},
     year = {2007},
     note = {talk:961},
     pages = {237-294},
     zbl = {1194.46085},
     mrnumber = {2359046},
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     url = {http://www.numdam.org/item/SB_2005-2006__48__237_0}
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Vaes, Stefaan. Rigidity results for Bernoulli actions and their von Neumann algebras, in Séminaire Bourbaki : volume 2005/2006, exposés 952-966, Astérisque, no. 311 (2007), Talk no. 961, pp. 237-294. http://www.numdam.org/item/SB_2005-2006__48__237_0/

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