Explicit computations of all finite index bimodules for a family of II$_1$ factors

Vaes, Stefaan

Explicit computations of all finite index bimodules for a family of II

_{1}

factors
[Calculs explicites de tous les bimodules d’indice infini d’une famille de facteurs de type II

_{1}

]

Vaes, Stefaan

Annales scientifiques de l'École Normale Supérieure, Tome 41 (2008), p. 743-788 / Harvested from Numdam

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Résumé
Abstract

Nous étudions des facteurs $M$ et $N$ de type II $_{1}$ associés à de bonnes actions Bernoulli généralisées de groupes $Γ$ et $Λ$ ayant un sous-groupe infini presque-distingué avec la propriété (T) relative. Nous démontrons le résultat de rigidité suivant : chaque $M$ - $N$ -bimodule d’indice fini (en particulier, chaque isomorphisme entre $M$ et $N$ ) peut être décrit par une commensurabilité des groupes $Γ$ , $Λ$ et une commensurabilité de leurs actions. L’algèbre de fusion des $M$ - $M$ -bimodules d’indice fini est identifiée avec une algèbre de Hecke étendue, ce qui fournit les premiers calculs explicites de l’algèbre de fusion d’un facteur de type II $_{1}$ . Nous obtenons en particulier des exemples explicites de facteurs II $_{1}$ dont l’algèbre de fusion est triviale, ce qui veut dire que tous leurs sous-facteurs d’indice fini sont triviaux.

We study II $_{1}$ factors $M$ and $N$ associated with good generalized Bernoulli actions of groups having an infinite almost normal subgroup with the relative property (T). We prove the following rigidity result : every finite index $M$ - $N$ -bimodule (in particular, every isomorphism between $M$ and $N$ ) is described by a commensurability of the groups involved and a commensurability of their actions. The fusion algebra of finite index $M$ - $M$ -bimodules is identified with an extended Hecke fusion algebra, providing the first explicit computations of the fusion algebra of a II $_{1}$ factor. We obtain in particular explicit examples of II $_{1}$ factors with trivial fusion algebra, i.e. only having trivial finite index subfactors.

Publié le : 2008-01-01

MR 2504433 | Zbl 1194.46086 | 1 citation dans Numdam

DOI : https://doi.org/10.24033/asens.2081

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     author = {Vaes, Stefaan},
     title = {Explicit computations of all finite index bimodules for a family of II$\_1$ factors},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     volume = {41},
     year = {2008},
     pages = {743-788},
     doi = {10.24033/asens.2081},
     mrnumber = {2504433},
     zbl = {1194.46086},
     language = {en},
     url = {http://dml.mathdoc.fr/item/ASENS_2008_4_41_5_743_0}
}

Vaes, Stefaan. Explicit computations of all finite index bimodules for a family of II$_1$ factors. Annales scientifiques de l'École Normale Supérieure, Tome 41 (2008) pp. 743-788. doi : 10.24033/asens.2081. http://gdmltest.u-ga.fr/item/ASENS_2008_4_41_5_743_0/

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