Explicit computations of all finite index bimodules for a family of II 1 factors

Stefaan Vaes

Annales scientifiques de l'École Normale Supérieure (2008)

  • Volume: 41, Issue: 5, page 743-788
  • ISSN: 0012-9593

Abstract

top
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.

How to cite

top

Vaes, Stefaan. "Explicit computations of all finite index bimodules for a family of II$_1$ factors." Annales scientifiques de l'École Normale Supérieure 41.5 (2008): 743-788. <http://eudml.org/doc/272130>.

@article{Vaes2008,
abstract = {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.},
author = {Vaes, Stefaan},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {generalized Bernoulli actions; good actions of good groups; II factors; fusion algebra; outer automorphism group; minimal condition on stabilizers},
language = {eng},
number = {5},
pages = {743-788},
publisher = {Société mathématique de France},
title = {Explicit computations of all finite index bimodules for a family of II$_1$ factors},
url = {http://eudml.org/doc/272130},
volume = {41},
year = {2008},
}

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
PB - Société mathématique de France
VL - 41
IS - 5
SP - 743
EP - 788
AB - 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.
LA - eng
KW - generalized Bernoulli actions; good actions of good groups; II factors; fusion algebra; outer automorphism group; minimal condition on stabilizers
UR - http://eudml.org/doc/272130
ER -

References

top
  1. [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. Zbl0202.03201MR259020
  2. [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. Zbl0842.46040
  3. [3] R. M. Bryant, Groups with the minimal condition on centralizers, J. Algebra60 (1979), 371–383. Zbl0422.20022MR549936
  4. [4] I. Bumagin & D. T. Wise, Every group is an outer automorphism group of a finitely generated group, J. Pure Appl. Algebra200 (2005), 137–147. Zbl1082.20021
  5. [5] A. Connes, A factor of type II 1 with countable fundamental group, J. Operator Theory4 (1980), 151–153. Zbl0455.46056MR587372
  6. [6] A. Connes, Noncommutative geometry, Academic Press Inc., 1994. Zbl0818.46076MR1303779
  7. [7] A. Connes, J. Feldman & B. Weiss, An amenable equivalence relation is generated by a single transformation, Ergodic Theory Dynamical Systems1 (1981), 431–450. Zbl0491.28018
  8. [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. Zbl1118.20030
  9. [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. Zbl1153.46036
  10. [10] A. Furman, On Popa’s cocycle superrigidity theorem, Int. Math. Res. Not. IMRN (2007), Art. ID rnm073. Zbl1134.46043MR2359545
  11. [11] C. Houdayer, Construction of type II 1 factors with prescribed countable fundamental group, arXiv:0704.3502, to appear in J. reine angew. Math. Zbl1209.46038MR2560409
  12. [12] A. Ioana, Rigidity results for wreath product II 1 factors, J. Funct. Anal.252 (2007), 763–791. Zbl1134.46041MR2360936
  13. [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. Zbl1149.46047
  14. [14] V. F. R. Jones, Index for subfactors, Invent. Math.72 (1983), 1–25. Zbl0508.46040MR696688
  15. [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. Zbl0831.46060
  16. [16] F. J. Murray & J. Von Neumann, On rings of operators, Ann. of Math.37 (1936), 116–229. Zbl0014.16101
  17. [17] J. v. Neumann & E. P. Wigner, Minimally almost periodic groups, Ann. of Math. 41 (1940), 746–750. Zbl0025.10106JFM66.0544.02
  18. [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. Zbl0427.28018
  19. [19] M. Pimsner & S. Popa, Entropy and index for subfactors, Ann. Sci. École Norm. Sup.19 (1986), 57–106. Zbl0646.46057
  20. [20] S. Popa, Correspondences, INCREST preprint http://www.math.ucla.edu/~popa/popa-correspondences.pdf, 1986. 
  21. [21] S. Popa, On a class of type II 1 factors with Betti numbers invariants, Ann. of Math.163 (2006), 809–899. Zbl1120.46045MR2215135
  22. [22] S. Popa, Strong rigidity of II 1 factors arising from malleable actions of w -rigid groups I, Invent. Math.165 (2006), 369–408. Zbl1120.46043MR2231961
  23. [23] S. Popa, Strong rigidity of II 1 factors arising from malleable actions of w -rigid groups II, Invent. Math.165 (2006), 409–451. Zbl1120.46044MR2231962
  24. [24] S. Popa, Cocycle and orbit equivalence superrigidity for malleable actions of w -rigid groups, Invent. Math.170 (2007), 243–295. Zbl1131.46040MR2342637
  25. [25] S. Popa & S. Vaes, Strong rigidity of generalized Bernoulli actions and computations of their symmetry groups, Adv. Math.217 (2008), 833–872. Zbl1137.37003
  26. [26] O. Schreier & B. L. v. d. Waerden, Die Automorphismen der projektiven Gruppen, Abhandlungen Hamburg 6 (1928), 303–322. JFM54.0149.02
  27. [27] B. Truffault, Centralisateurs des éléments dans les groupes de Greendlinger, C. R. Acad. Sci. Paris279 (1974), 317–319. Zbl0291.20040MR384943
  28. [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. Zbl1194.46085MR2359046
  29. [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. Zbl1172.46043

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.