Inventiones mathematicae

, Volume 182, Issue 2, pp 371–417

Group measure space decomposition of II1 factors and W*-superrigidity



We prove a “unique crossed product decomposition” result for group measure space II1 factors L (X)⋊Γ arising from arbitrary free ergodic probability measure preserving (p.m.p.) actions of groups Γ in a fairly large family \(\mathcal{G}\), which contains all free products of a Kazhdan group and a non-trivial group, as well as certain amalgamated free products over an amenable subgroup. We deduce that if Tn denotes the group of upper triangular matrices in PSL (n,ℤ), then any free, mixing p.m.p. action of \(\Gamma=\operatorname{PSL}(n,\mathbb{Z})*_{T_{n}}\operatorname{PSL}(n,\mathbb{Z})\) is W-superrigid, i.e. any isomorphism between L (X)⋊Γ and an arbitrary group measure space factor L (Y)⋊Λ, comes from a conjugacy of the actions. We also prove that for many groups Γ in the family \(\mathcal{G}\), the Bernoulli actions of Γ are W-superrigid.


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  1. 1.
    Anantharaman-Delaroche, C.: Amenable correspondences and approximation properties for von Neumann algebras. Pac. J. Math. 171, 309–341 (1995) MATHMathSciNetGoogle Scholar
  2. 2.
    Bowen, L.: Orbit equivalence, coinduced actions and free products. Groups Geom. Dyn. (2010, to appear). arXiv:0906.4573
  3. 3.
    Bożejko, M., Picardello, M.A.: Weakly amenable groups and amalgamated products. Proc. Am. Math. Soc. 117, 1039–1046 (1993) MATHGoogle Scholar
  4. 4.
    Chifan, I., Houdayer, C.: Bass-Serre rigidity results in von Neumann algebras. Duke Math. J. 153, 23–54 (2010) MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Connes, A.: Sur la classification des facteurs de type II. C. R. Acad. Sci. Paris 281, 13–15 (1975) MATHMathSciNetGoogle Scholar
  6. 6.
    Connes, A.: Classification of injective factors. Ann. Math. (2) 104, 73–115 (1976) CrossRefMathSciNetGoogle Scholar
  7. 7.
    Connes, A.: A factor of type II1 with countable fundamental group. J. Oper. Theory 4, 151–153 (1980) MATHMathSciNetGoogle Scholar
  8. 8.
    Connes, A.: Classification des facteurs. In: Operator Algebras and Applications, Part 2, Kingston, 1980. Proc. Sympos. Pure Math., vol. 38, pp. 43–109. Am. Math. Soc., Providence (1982) Google Scholar
  9. 9.
    Connes, A.: Noncommutative Geometry. Academic Press, New York (1994) MATHGoogle Scholar
  10. 10.
    Connes, A., Jones, V.F.R.: A II1 factor with two non-conjugate Cartan subalgebras. Bull. Am. Math. Soc. 6, 211–212 (1982) MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Connes, A., Jones, V.F.R.: Property (T) for von Neumann algebras. Bull. Lond. Math. Soc. 17, 57–62 (1985) MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Connes, A., Feldman, J., Weiss, B.: An amenable equivalence relation is generated by a single transformation. Ergod. Theory Dyn. Syst. 1, 431–450 (1981) MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Cornulier, Y., Stalder, Y., Valette, A.: Proper actions of wreath products and generalizations. Preprint, arXiv:0905.3960
  14. 14.
    Cowling, M., Haagerup, U.: Completely bounded multipliers of the Fourier algebra of a simple Lie group of real rank one. Invent. Math. 96, 507–549 (1989) MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Dye, H.A.: On groups of measure preserving transformations. I. Am. J. Math. 81, 119–159 (1959) MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Dye, H.A.: On groups of measure preserving transformations. II. Am. J. Math. 85, 551–576 (1963) MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Feldman, J., Moore, C.C.: Ergodic equivalence relations, cohomology, and von Neumann algebras, II. Trans. Am. Math. Soc. 234, 325–359 (1977) MATHMathSciNetGoogle Scholar
  18. 18.
    Furman, A.: Gromov’s measure equivalence and rigidity of higher rank lattices. Ann. Math. 150, 1059–1081 (1999) MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Furman, A.: Orbit equivalence rigidity. Ann. Math. 150, 1083–1108 (1999) MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Furman, A.: On Popa’s cocycle superrigidity theorem. Int. Math. Res. Not. (2007), Art. ID rnm073, 46 pp. Google Scholar
  21. 21.
    Gaboriau, D.: Coût des relations d’équivalence et des groupes. Invent. Math. 139, 41–98 (2000) MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Gaboriau, D.: Invariants 2 de relations d’équivalence et de groupes. Publ. Math. Inst. Hautes Études Sci. 95, 93–150 (2002) MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Gaboriau, D.: Examples of groups that are measure equivalent to the free group. Ergod. Theory Dyn. Syst. 25, 1809–1827 (2005) MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Hjorth, G.: A converse to Dye’s theorem. Trans. Am. Math. Soc. 357, 3083–3103 (2004) CrossRefMathSciNetGoogle Scholar
  25. 25.
    Hjorth, G., Kechris, A.: Rigidity theorems for actions of product groups and countable Borel equivalence relations. Mem. Am. Math. Soc. 177(833) (2005) Google Scholar
  26. 26.
    Houdayer, C.: Construction of type II1 factors with prescribed countable fundamental group. J. Reine Angew Math. 634, 169–207 (2009) MATHMathSciNetGoogle Scholar
  27. 27.
    Ioana, A.: Cocycle superrigidity for profinite actions of property (T) groups. Duke Math. J., to appear. arXiv:0805.2998
  28. 28.
    Ioana, A., Peterson, J., Popa, S.: Amalgamated free products of weakly rigid factors and calculation of their symmetry groups. Acta Math. 200, 85–153 (2008) MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Jones, V.F.R.: Index for subfactors. Invent. Math. 72, 1–25 (1983) MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Jones, V.F.R.: Ten problems. In: Mathematics: Frontiers and Perspectives. Am. Math. Soc., Providence (2000), pp. 79–91 Google Scholar
  31. 31.
    Kida, Y.: Measure equivalence rigidity of the mapping class group. Ann. Math. 171, 1851–1901 (2010) MATHCrossRefGoogle Scholar
  32. 32.
    Kida, Y.: Rigidity of amalgamated free products in measure equivalence theory. Preprint, arXiv:0902.2888
  33. 33.
    McDuff, D.: Central sequences and the hyperfinite factor. Proc. Lond. Math. Soc. (3) 21, 443–461 (1970) MATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    Monod, N., Shalom, Y.: Orbit equivalence rigidity and bounded cohomology. Ann. Math. 164, 825–878 (2006) MATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    Murray, F.J., von Neumann, J.: On rings of operators. Ann. Math. 37, 116–229 (1936) CrossRefGoogle Scholar
  36. 36.
    Murray, F.J., von Neumann, J.: Rings of operators IV. Ann. Math. 44, 716–808 (1943) CrossRefGoogle Scholar
  37. 37.
    Ornstein, D.S., Weiss, B.: Ergodic theory of amenable group actions. Bull. Am. Math. Soc. (N.S.) 2, 161–164 (1980) MATHCrossRefMathSciNetGoogle Scholar
  38. 38.
    Ozawa, N.: Solid von Neumann algebras. Acta Math. 192, 111–117 (2004) MATHCrossRefMathSciNetGoogle Scholar
  39. 39.
    Ozawa, N., Popa, S.: On a class of II1 factors with at most one Cartan subalgebra. Ann. Math. 172, 713–749 (2010) MATHCrossRefGoogle Scholar
  40. 40.
    Ozawa, N., Popa, S.: On a class of II1 factors with at most one Cartan subalgebra, II. Am. J. Math. 132, 841–866 (2010) MATHCrossRefMathSciNetGoogle Scholar
  41. 41.
    Peterson, J.: L 2-rigidity in von Neumann algebras. Invent. Math. 175, 417–433 (2009) MATHCrossRefMathSciNetGoogle Scholar
  42. 42.
    Peterson, J.: Examples of group actions which are virtually W-superrigid. Preprint, arXiv:1002.1745
  43. 43.
    Popa, S.: Markov traces on universal Jones algebras and subfactors of finite index. Invent. Math. 111, 375–405 (1993) MATHCrossRefMathSciNetGoogle Scholar
  44. 44.
    Popa, S.: On a class of type II1 factors with Betti numbers invariants. Ann. Math. 163, 809–899 (2006) MATHCrossRefGoogle Scholar
  45. 45.
    Popa, S.: Strong rigidity of II1 factors arising from malleable actions of w-rigid groups, Part I. Invent. Math. 165, 369–408 (2006) MATHCrossRefMathSciNetGoogle Scholar
  46. 46.
    Popa, S.: Strong rigidity of II1 factors arising from malleable actions of w-rigid groups, II. Invent. Math. 165, 409–452 (2006) MATHCrossRefMathSciNetGoogle Scholar
  47. 47.
    Popa, S.: Cocycle and orbit equivalence superrigidity for malleable actions of w-rigid groups. Invent. Math. 170, 243–295 (2007) MATHCrossRefMathSciNetGoogle Scholar
  48. 48.
    Popa, S.: Deformation and rigidity for group actions and von Neumann algebras. In: Proceedings of the International Congress of Mathematicians. vol. I. Madrid, 2006, pp. 445–477. European Mathematical Society Publishing House, Zürich (2007) CrossRefGoogle Scholar
  49. 49.
    Popa, S.: On the superrigidity of malleable actions with spectral gap. J. Am. Math. Soc. 21, 981–1000 (2008) CrossRefGoogle Scholar
  50. 50.
    Popa, S., Sasyk, R.: On the cohomology of Bernoulli actions. Ergod. Theory Dyn. Syst. 27, 241–251 (2007) MATHCrossRefMathSciNetGoogle Scholar
  51. 51.
    Popa, S., Vaes, S.: Strong rigidity of generalized Bernoulli actions and computations of their symmetry groups. Adv. Math. 217, 833–872 (2008) MATHCrossRefMathSciNetGoogle Scholar
  52. 52.
    Popa, S., Vaes, S.: Actions of \(\mathbb{F}_{\infty}\) whose II1 factors and orbit equivalence relations have prescribed fundamental group. J. Am. Math. Soc. 23, 383–403 (2010) MATHCrossRefMathSciNetGoogle Scholar
  53. 53.
    Popa, S., Vaes, S.: Cocycle and orbit superrigidity for lattices in SL (n,ℝ) acting on homogeneous spaces. In: Geometry, Rigidity and Group Actions. Proceedings of the Conference in Honor of R.J. Zimmer’s 60th Birthday (2010, to appear). arXiv:0810.3630
  54. 54.
    Popa, S., Vaes, S.: On the fundamental group of II1 factors and equivalence relations arising from group actions. In: Quanta of Maths. Proceedings of the Conference in Honor of A. Connes’ 60th Birthday (2010, to appear). arXiv:0810.0706
  55. 55.
    Singer, I.M.: Automorphisms of finite factors. Am. J. Math. 77, 117–133 (1955) MATHCrossRefGoogle Scholar
  56. 56.
    Stepin, A.M.: Bernoulli shifts on groups and decreasing sequences of partitions. In: Proceedings of the Third Japan–USSR Symposium on Probability Theory, Tashkent, 1975. Lecture Notes in Math., vol. 550, pp. 592–603. Springer, Berlin (1976) CrossRefGoogle Scholar
  57. 57.
    Vaes, S.: Rigidity results for Bernoulli actions and their von Neumann algebras (after Sorin Popa). Séminaire Bourbaki, exp. no. 961. Astérisque 311, 237–294 (2007) MathSciNetGoogle Scholar
  58. 58.
    Vaes, S.: Explicit computations of all finite index bimodules for a family of II1 factors. Ann. Sci. École Norm. Super. 41, 743–788 (2008) MATHMathSciNetGoogle Scholar
  59. 59.
    Voiculescu, D.V., Dykema, K.J., Nica, A.: Free Random Variables. CRM Monograph Series, vol. 1. American Mathematical Society, Providence (1992) MATHGoogle Scholar
  60. 60.
    Zimmer, R.J.: Strong rigidity for ergodic actions of semisimple Lie groups. Ann. Math. 112, 511–529 (1980) CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Mathematics DepartmentUniversity of California at Los AngelesLos AngelesUSA
  2. 2.Department of MathematicsK.U. LeuvenLeuvenBelgium

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