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

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Abstract

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 T n 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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References

  1. Anantharaman-Delaroche, C.: Amenable correspondences and approximation properties for von Neumann algebras. Pac. J. Math. 171, 309–341 (1995)

    MATH  MathSciNet  Google Scholar 

  2. Bowen, L.: Orbit equivalence, coinduced actions and free products. Groups Geom. Dyn. (2010, to appear). arXiv:0906.4573

  3. Bożejko, M., Picardello, M.A.: Weakly amenable groups and amalgamated products. Proc. Am. Math. Soc. 117, 1039–1046 (1993)

    MATH  Google Scholar 

  4. Chifan, I., Houdayer, C.: Bass-Serre rigidity results in von Neumann algebras. Duke Math. J. 153, 23–54 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  5. Connes, A.: Sur la classification des facteurs de type II. C. R. Acad. Sci. Paris 281, 13–15 (1975)

    MATH  MathSciNet  Google Scholar 

  6. Connes, A.: Classification of injective factors. Ann. Math. (2) 104, 73–115 (1976)

    Article  MathSciNet  Google Scholar 

  7. Connes, A.: A factor of type II1 with countable fundamental group. J. Oper. Theory 4, 151–153 (1980)

    MATH  MathSciNet  Google Scholar 

  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. Connes, A.: Noncommutative Geometry. Academic Press, New York (1994)

    MATH  Google Scholar 

  10. Connes, A., Jones, V.F.R.: A II1 factor with two non-conjugate Cartan subalgebras. Bull. Am. Math. Soc. 6, 211–212 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  11. Connes, A., Jones, V.F.R.: Property (T) for von Neumann algebras. Bull. Lond. Math. Soc. 17, 57–62 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  13. Cornulier, Y., Stalder, Y., Valette, A.: Proper actions of wreath products and generalizations. Preprint, arXiv:0905.3960

  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)

    Article  MATH  MathSciNet  Google Scholar 

  15. Dye, H.A.: On groups of measure preserving transformations. I. Am. J. Math. 81, 119–159 (1959)

    Article  MATH  MathSciNet  Google Scholar 

  16. Dye, H.A.: On groups of measure preserving transformations. II. Am. J. Math. 85, 551–576 (1963)

    Article  MATH  MathSciNet  Google Scholar 

  17. Feldman, J., Moore, C.C.: Ergodic equivalence relations, cohomology, and von Neumann algebras, II. Trans. Am. Math. Soc. 234, 325–359 (1977)

    MATH  MathSciNet  Google Scholar 

  18. Furman, A.: Gromov’s measure equivalence and rigidity of higher rank lattices. Ann. Math. 150, 1059–1081 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  19. Furman, A.: Orbit equivalence rigidity. Ann. Math. 150, 1083–1108 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  20. Furman, A.: On Popa’s cocycle superrigidity theorem. Int. Math. Res. Not. (2007), Art. ID rnm073, 46 pp.

  21. Gaboriau, D.: Coût des relations d’équivalence et des groupes. Invent. Math. 139, 41–98 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  22. Gaboriau, D.: Invariants ℓ 2 de relations d’équivalence et de groupes. Publ. Math. Inst. Hautes Études Sci. 95, 93–150 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  23. Gaboriau, D.: Examples of groups that are measure equivalent to the free group. Ergod. Theory Dyn. Syst. 25, 1809–1827 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  24. Hjorth, G.: A converse to Dye’s theorem. Trans. Am. Math. Soc. 357, 3083–3103 (2004)

    Article  MathSciNet  Google Scholar 

  25. Hjorth, G., Kechris, A.: Rigidity theorems for actions of product groups and countable Borel equivalence relations. Mem. Am. Math. Soc. 177(833) (2005)

  26. Houdayer, C.: Construction of type II1 factors with prescribed countable fundamental group. J. Reine Angew Math. 634, 169–207 (2009)

    MATH  MathSciNet  Google Scholar 

  27. Ioana, A.: Cocycle superrigidity for profinite actions of property (T) groups. Duke Math. J., to appear. arXiv:0805.2998

  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)

    Article  MATH  MathSciNet  Google Scholar 

  29. Jones, V.F.R.: Index for subfactors. Invent. Math. 72, 1–25 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  30. Jones, V.F.R.: Ten problems. In: Mathematics: Frontiers and Perspectives. Am. Math. Soc., Providence (2000), pp. 79–91

    Google Scholar 

  31. Kida, Y.: Measure equivalence rigidity of the mapping class group. Ann. Math. 171, 1851–1901 (2010)

    Article  MATH  Google Scholar 

  32. Kida, Y.: Rigidity of amalgamated free products in measure equivalence theory. Preprint, arXiv:0902.2888

  33. McDuff, D.: Central sequences and the hyperfinite factor. Proc. Lond. Math. Soc. (3) 21, 443–461 (1970)

    Article  MATH  MathSciNet  Google Scholar 

  34. Monod, N., Shalom, Y.: Orbit equivalence rigidity and bounded cohomology. Ann. Math. 164, 825–878 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  35. Murray, F.J., von Neumann, J.: On rings of operators. Ann. Math. 37, 116–229 (1936)

    Article  Google Scholar 

  36. Murray, F.J., von Neumann, J.: Rings of operators IV. Ann. Math. 44, 716–808 (1943)

    Article  Google Scholar 

  37. Ornstein, D.S., Weiss, B.: Ergodic theory of amenable group actions. Bull. Am. Math. Soc. (N.S.) 2, 161–164 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  38. Ozawa, N.: Solid von Neumann algebras. Acta Math. 192, 111–117 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  39. Ozawa, N., Popa, S.: On a class of II1 factors with at most one Cartan subalgebra. Ann. Math. 172, 713–749 (2010)

    Article  MATH  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  41. Peterson, J.: L 2-rigidity in von Neumann algebras. Invent. Math. 175, 417–433 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  42. Peterson, J.: Examples of group actions which are virtually W∗-superrigid. Preprint, arXiv:1002.1745

  43. Popa, S.: Markov traces on universal Jones algebras and subfactors of finite index. Invent. Math. 111, 375–405 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  44. Popa, S.: On a class of type II1 factors with Betti numbers invariants. Ann. Math. 163, 809–899 (2006)

    Article  MATH  Google Scholar 

  45. Popa, S.: Strong rigidity of II1 factors arising from malleable actions of w-rigid groups, Part I. Invent. Math. 165, 369–408 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  46. Popa, S.: Strong rigidity of II1 factors arising from malleable actions of w-rigid groups, II. Invent. Math. 165, 409–452 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  47. Popa, S.: Cocycle and orbit equivalence superrigidity for malleable actions of w-rigid groups. Invent. Math. 170, 243–295 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Chapter  Google Scholar 

  49. Popa, S.: On the superrigidity of malleable actions with spectral gap. J. Am. Math. Soc. 21, 981–1000 (2008)

    Article  Google Scholar 

  50. Popa, S., Sasyk, R.: On the cohomology of Bernoulli actions. Ergod. Theory Dyn. Syst. 27, 241–251 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  51. Popa, S., Vaes, S.: Strong rigidity of generalized Bernoulli actions and computations of their symmetry groups. Adv. Math. 217, 833–872 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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. 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. Singer, I.M.: Automorphisms of finite factors. Am. J. Math. 77, 117–133 (1955)

    Article  MATH  Google Scholar 

  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)

    Chapter  Google Scholar 

  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)

    MathSciNet  Google Scholar 

  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)

    MATH  MathSciNet  Google Scholar 

  59. Voiculescu, D.V., Dykema, K.J., Nica, A.: Free Random Variables. CRM Monograph Series, vol. 1. American Mathematical Society, Providence (1992)

    MATH  Google Scholar 

  60. Zimmer, R.J.: Strong rigidity for ergodic actions of semisimple Lie groups. Ann. Math. 112, 511–529 (1980)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Mathematics Department, University of California at Los Angeles, Los Angeles, CA, 90095-1555, USA

    Sorin Popa

  2. Department of Mathematics, K.U. Leuven, Celestijnenlaan 200B, 3001, Leuven, Belgium

    Stefaan Vaes

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  1. Sorin Popa
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  2. Stefaan Vaes
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Correspondence to Stefaan Vaes.

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S. Popa was partially supported by NSF Grant DMS-0601082.

S. Vaes was partially supported by ERC Starting Grant VNALG-200749, Research Programme G.0231.07 of the Research Foundation—Flanders (FWO) and K.U. Leuven BOF research Grant OT/08/032.

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Popa, S., Vaes, S. Group measure space decomposition of II1 factors and W*-superrigidity. Invent. math. 182, 371–417 (2010). https://doi.org/10.1007/s00222-010-0268-5

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