We study II factors and 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 --bimodule (in particular, every isomorphism between and ) is described by a commensurability of the groups involved and a commensurability of their actions. The fusion algebra of finite index --bimodules is identified with an extended Hecke fusion algebra, providing the first explicit computations of the fusion algebra of a II factor. We obtain in particular explicit examples of II factors with trivial fusion algebra, i.e. only having trivial finite index subfactors.
Nous étudions des facteurs et de type II associés à de bonnes actions Bernoulli généralisées de groupes et ayant un sous-groupe infini presque-distingué avec la propriété (T) relative. Nous démontrons le résultat de rigidité suivant : chaque --bimodule d’indice fini (en particulier, chaque isomorphisme entre et ) peut être décrit par une commensurabilité des groupes , et une commensurabilité de leurs actions. L’algèbre de fusion des --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. Nous obtenons en particulier des exemples explicites de facteurs II dont l’algèbre de fusion est triviale, ce qui veut dire que tous leurs sous-facteurs d’indice fini sont triviaux.
@article{ASENS_2008_4_41_5_743_0, 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}, language = {en}, url = {http://www.numdam.org/articles/10.24033/asens.2081/} }
TY - JOUR AU - Vaes, Stefaan TI - Explicit computations of all finite index bimodules for a family of II$_1$ factors JO - Annales scientifiques de l'École Normale Supérieure PY - 2008 SP - 743 EP - 788 VL - 41 IS - 5 PB - Société mathématique de France UR - http://www.numdam.org/articles/10.24033/asens.2081/ DO - 10.24033/asens.2081 LA - en ID - ASENS_2008_4_41_5_743_0 ER -
%0 Journal Article %A Vaes, Stefaan %T Explicit computations of all finite index bimodules for a family of II$_1$ factors %J Annales scientifiques de l'École Normale Supérieure %D 2008 %P 743-788 %V 41 %N 5 %I Société mathématique de France %U http://www.numdam.org/articles/10.24033/asens.2081/ %R 10.24033/asens.2081 %G en %F 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, Serie 4, Volume 41 (2008) no. 5, pp. 743-788. doi : 10.24033/asens.2081. http://www.numdam.org/articles/10.24033/asens.2081/
[1] On the automorphisms of certain subgroups of semi-simple Lie groups., in Algebraic Geometry (Internat. Colloq., Tata Inst. Fund. Res., Bombay, 1968), Oxford Univ. Press, 1969, 43-73. | MR | Zbl
,[2] Hecke algebras, type III factors and phase transitions with spontaneous symmetry breaking in number theory, Selecta Math. (N.S.) 1 (1995), 411-457. | Zbl
& ,[3] Groups with the minimal condition on centralizers, J. Algebra 60 (1979), 371-383. | MR | Zbl
,[4] Every group is an outer automorphism group of a finitely generated group, J. Pure Appl. Algebra 200 (2005), 137-147. | Zbl
& ,[5] A factor of type with countable fundamental group, J. Operator Theory 4 (1980), 151-153. | MR | Zbl
,[6] Noncommutative geometry, Academic Press Inc., 1994. | MR | Zbl
,[7] An amenable equivalence relation is generated by a single transformation, Ergodic Theory Dynamical Systems 1 (1981), 431-450. | Zbl
, & ,[8] Centraliser dimension and universal classes of groups, Sib. Èlektron. Mat. Izv. 3 (2006), 197-215, arXiv:math/0502498. | Zbl
, & ,[9] Every compact group arises as the outer automorphism group of a factor, J. Funct. Anal. 254 (2008), 2317-2328. | Zbl
& ,[10] On Popa's cocycle superrigidity theorem, Int. Math. Res. Not. IMRN (2007), Art. ID rnm073. | MR | Zbl
,[11] Construction of type II factors with prescribed countable fundamental group, arXiv:0704.3502, to appear in J. reine angew. Math. | MR | Zbl
,[12] Rigidity results for wreath product factors, J. Funct. Anal. 252 (2007), 763-791. | MR | Zbl
,[13] Amalgamated free products of weakly rigid factors and calculation of their symmetry groups, Acta Math. 200 (2008), 85-153. | Zbl
, & ,[14] Index for subfactors, Invent. Math. 72 (1983), 1-25. | MR | Zbl
,[15] Fundamentals of the theory of operator algebras. Vol. II, Pure and Appl. Math. 100, Academic Press Inc., 1986. | Zbl
& ,[16] On rings of operators, Ann. of Math. 37 (1936), 116-229. | Zbl
& ,[17] J. v. Neumann & E. P. Wigner, Minimally almost periodic groups, Ann. of Math. 41 (1940), 746-750. | JFM | Zbl
[18] Ergodic theory of amenable group actions. I. The Rohlin lemma, Bull. Amer. Math. Soc. (N.S.) 2 (1980), 161-164. | Zbl
& ,[19] Entropy and index for subfactors, Ann. Sci. École Norm. Sup. 19 (1986), 57-106. | Numdam | Zbl
& ,[20] Correspondences, INCREST preprint http://www.math.ucla.edu/~popa/popa-correspondences.pdf, 1986.
,[21] On a class of type factors with Betti numbers invariants, Ann. of Math. 163 (2006), 809-899. | MR | Zbl
,[22] Strong rigidity of factors arising from malleable actions of -rigid groups I, Invent. Math. 165 (2006), 369-408. | MR | Zbl
,[23] Strong rigidity of factors arising from malleable actions of -rigid groups II, Invent. Math. 165 (2006), 409-451. | MR | Zbl
,[24] Cocycle and orbit equivalence superrigidity for malleable actions of -rigid groups, Invent. Math. 170 (2007), 243-295. | MR | Zbl
,[25] Strong rigidity of generalized Bernoulli actions and computations of their symmetry groups, Adv. Math. 217 (2008), 833-872. | Zbl
& ,[26] O. Schreier & B. L. v. d. Waerden, Die Automorphismen der projektiven Gruppen, Abhandlungen Hamburg 6 (1928), 303-322. | JFM
[27] Centralisateurs des éléments dans les groupes de Greendlinger, C. R. Acad. Sci. Paris 279 (1974), 317-319. | MR | Zbl
,[28] Rigidity results for Bernoulli actions and their von Neumann algebras (after Sorin Popa), Sém. Bourbaki, vol. 2005/2006, exposé no 961, Astérisque 311 (2007), 237-294. | Numdam | MR | Zbl
,[29] Factors of type II without non-trivial finite index subfactors, Trans. of the AMS, in print. DOI: 10.1090/S0002-9947-08-04585-6. | Zbl
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