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, p. 743-788

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.

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.

@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},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {Ser. 4, 41},
     number = {5},
     year = {2008},
     pages = {743-788},
     doi = {10.24033/asens.2081},
     zbl = {1194.46086},
     mrnumber = {2504433},
     language = {en},
     url = {http://www.numdam.org/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, Serie 4, Volume 41 (2008) no. 5, pp. 743-788. doi : 10.24033/asens.2081. http://www.numdam.org/item/ASENS_2008_4_41_5_743_0/

[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 259020 | Zbl 0202.03201

[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 0842.46040

[3] R. M. Bryant, Groups with the minimal condition on centralizers, J. Algebra 60 (1979), 371-383. | MR 549936 | Zbl 0422.20022

[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 1082.20021

[5] A. Connes, A factor of type II 1 with countable fundamental group, J. Operator Theory 4 (1980), 151-153. | MR 587372 | Zbl 0455.46056

[6] A. Connes, Noncommutative geometry, Academic Press Inc., 1994. | MR 1303779 | Zbl 0818.46076

[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 0491.28018

[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 1118.20030

[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 1153.46036

[10] A. Furman, On Popa's cocycle superrigidity theorem, Int. Math. Res. Not. IMRN (2007), Art. ID rnm073. | MR 2359545 | Zbl 1134.46043

[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 2560409 | Zbl 1209.46038

[12] A. Ioana, Rigidity results for wreath product II 1 factors, J. Funct. Anal. 252 (2007), 763-791. | MR 2360936 | Zbl 1134.46041

[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 1149.46047

[14] V. F. R. Jones, Index for subfactors, Invent. Math. 72 (1983), 1-25. | MR 696688 | Zbl 0508.46040

[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 0831.46060

[16] F. J. Murray & J. Von Neumann, On rings of operators, Ann. of Math. 37 (1936), 116-229. | Zbl 0014.16101

[17] J. v. Neumann & E. P. Wigner, Minimally almost periodic groups, Ann. of Math. 41 (1940), 746-750. | JFM 66.0544.02 | Zbl 0025.10106

[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 0427.28018

[19] M. Pimsner & S. Popa, Entropy and index for subfactors, Ann. Sci. École Norm. Sup. 19 (1986), 57-106. | Numdam | Zbl 0646.46057

[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 2215135 | Zbl 1120.46045

[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 2231961 | Zbl 1120.46043

[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 2231962 | Zbl 1120.46044

[24] S. Popa, Cocycle and orbit equivalence superrigidity for malleable actions of w-rigid groups, Invent. Math. 170 (2007), 243-295. | MR 2342637 | Zbl 1131.46040

[25] S. Popa & S. Vaes, Strong rigidity of generalized Bernoulli actions and computations of their symmetry groups, Adv. Math. 217 (2008), 833-872. | Zbl 1137.37003

[26] O. Schreier & B. L. v. d. Waerden, Die Automorphismen der projektiven Gruppen, Abhandlungen Hamburg 6 (1928), 303-322. | JFM 54.0149.02

[27] B. Truffault, Centralisateurs des éléments dans les groupes de Greendlinger, C. R. Acad. Sci. Paris 279 (1974), 317-319. | MR 384943 | Zbl 0291.20040

[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 2359046 | Zbl 1194.46085

[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 1172.46043