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.

@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, Série 4, Tome 41 (2008) no. 5, pp. 743-788. doi : 10.24033/asens.2081. http://www.numdam.org/articles/10.24033/asens.2081/

[1] A. Borel, 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] J.-B. Bost & A. Connes, 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] R. M. Bryant, Groups with the minimal condition on centralizers, J. Algebra 60 (1979), 371-383. | MR | Zbl

[4] I. Bumagin & D. T. Wise, Every group is an outer automorphism group of a finitely generated group, J. Pure Appl. Algebra 200 (2005), 137-147. | Zbl

[5] A. Connes, A factor of type II 1 with countable fundamental group, J. Operator Theory 4 (1980), 151-153. | MR | Zbl

[6] A. Connes, Noncommutative geometry, Academic Press Inc., 1994. | MR | Zbl

[7] A. Connes, J. Feldman & B. Weiss, An amenable equivalence relation is generated by a single transformation, Ergodic Theory Dynamical Systems 1 (1981), 431-450. | Zbl

[8] A. J. Duncan, I. V. Kazachkov & V. N. Remeslennikov, Centraliser dimension and universal classes of groups, Sib. Èlektron. Mat. Izv. 3 (2006), 197-215, arXiv:math/0502498. | Zbl

[9] S. Falguières & S. Vaes, Every compact group arises as the outer automorphism group of a II 1 factor, J. Funct. Anal. 254 (2008), 2317-2328. | Zbl

[10] A. Furman, On Popa's cocycle superrigidity theorem, Int. Math. Res. Not. IMRN (2007), Art. ID rnm073. | MR | Zbl

[11] C. Houdayer, Construction of type II 1 factors with prescribed countable fundamental group, arXiv:0704.3502, to appear in J. reine angew. Math. | MR | Zbl

[12] A. Ioana, Rigidity results for wreath product II 1 factors, J. Funct. Anal. 252 (2007), 763-791. | MR | Zbl

[13] A. Ioana, J. Peterson & S. Popa, Amalgamated free products of weakly rigid factors and calculation of their symmetry groups, Acta Math. 200 (2008), 85-153. | Zbl

[14] V. F. R. Jones, Index for subfactors, Invent. Math. 72 (1983), 1-25. | MR | Zbl

[15] R. V. Kadison & J. R. Ringrose, Fundamentals of the theory of operator algebras. Vol. II, Pure and Appl. Math. 100, Academic Press Inc., 1986. | Zbl

[16] F. J. Murray & J. Von Neumann, 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] D. S. Ornstein & B. Weiss, Ergodic theory of amenable group actions. I. The Rohlin lemma, Bull. Amer. Math. Soc. (N.S.) 2 (1980), 161-164. | Zbl

[19] M. Pimsner & S. Popa, Entropy and index for subfactors, Ann. Sci. École Norm. Sup. 19 (1986), 57-106. | Numdam | Zbl

[20] S. Popa, Correspondences, INCREST preprint http://www.math.ucla.edu/~popa/popa-correspondences.pdf, 1986.

[21] S. Popa, On a class of type II 1 factors with Betti numbers invariants, Ann. of Math. 163 (2006), 809-899. | MR | Zbl

[22] S. Popa, Strong rigidity of II 1 factors arising from malleable actions of w-rigid groups I, Invent. Math. 165 (2006), 369-408. | MR | Zbl

[23] S. Popa, Strong rigidity of II 1 factors arising from malleable actions of w-rigid groups II, Invent. Math. 165 (2006), 409-451. | MR | Zbl

[24] S. Popa, Cocycle and orbit equivalence superrigidity for malleable actions of w-rigid groups, Invent. Math. 170 (2007), 243-295. | MR | Zbl

[25] S. Popa & S. Vaes, 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] B. Truffault, Centralisateurs des éléments dans les groupes de Greendlinger, C. R. Acad. Sci. Paris 279 (1974), 317-319. | MR | Zbl

[28] S. Vaes, 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] S. Vaes, Factors of type II 1 without non-trivial finite index subfactors, Trans. of the AMS, in print. DOI: 10.1090/S0002-9947-08-04585-6. | Zbl

Cité par Sources :