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 ]
Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 41 (2008) no. 5, pp. 743-788.

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.

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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},
     pages = {743--788},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {Ser. 4, 41},
     number = {5},
     year = {2008},
     doi = {10.24033/asens.2081},
     mrnumber = {2504433},
     zbl = {1194.46086},
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     url = {http://archive.numdam.org/articles/10.24033/asens.2081/}
}
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Vaes, Stefaan. Explicit computations of all finite index bimodules for a family of II$_1$ factors. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 41 (2008) no. 5, pp. 743-788. doi : 10.24033/asens.2081. http://archive.numdam.org/articles/10.24033/asens.2081/

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