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Rigidity results for von Neumann algebras arising from mixing extensions of profinite actions of groups on probability spaces

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Abstract

Motivated by Popa’s seminal work Popa (Invent Math 165:409-45, 2006), in this paper, we provide a fairly large class of examples of group actions \(\Gamma \curvearrowright X\) satisfying the extended Neshveyev–Størmer rigidity phenomenon Neshveyev and Størmer (J Funct Anal 195(2):239-261, 2002): whenever \(\Lambda \curvearrowright Y\) is a free ergodic pmp action and there is a \(*\)-isomorphism \(\Theta :L^\infty (X)\rtimes \Gamma {\rightarrow }L^\infty (Y)\rtimes \Lambda \) such that \(\Theta (L(\Gamma ))=L(\Lambda )\) then the actions \(\Gamma \curvearrowright X\) and \(\Lambda \curvearrowright Y\) are conjugate (in a way compatible with \(\Theta \)). We also obtain a complete description of the intermediate subalgebras of all (possibly non-free) compact extensions of group actions in the same spirit as the recent results of Suzuki (Complete descriptions of intermediate operator algebras by intermediate extensions of dynamical systems, To appear in Comm Math Phy. ArXiv Preprint: arXiv:1805.02077, 2020). This yields new consequences to the study of rigidity for crossed product von Neumann algebras and to the classification of subfactors of finite Jones index.

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Notes

  1. For every diffuse \(A\subseteq L(\Gamma )\) the relative commutant \(A'\cap L(\Gamma )\) is amenable

  2. For every diffuse amenable \(A\subseteq L(\Gamma )\) the normalizer \(\mathcal N_{L(\Gamma )}(A)''\) is amenable

References

  1. Alekseev, V., Brugger, R.: A rigidity result for normalized subfactors, Preprint arXiv:1903.04895

  2. Bratteli, O., Jørgensen, P.E.T., Kishimoto, A., Werner, R.F.: Pure states on \(O_d\). J. Operator Theory 43, 97–143 (2000)

    MathSciNet  MATH  Google Scholar 

  3. Bhattacharjee, M.: Constructing finitely presented infinite nearly simple groups. Comm. Algebra 22, 4561–4589 (1994)

    MathSciNet  MATH  Google Scholar 

  4. Boutonnet, R., Carderi, A.: Maximal amenable von Neumann subalgebras arising from maximal amenable subgroups. Geom. Funct. Anal. 25, 1688–1705 (2015)

    MathSciNet  MATH  Google Scholar 

  5. Boutonnet, R., Ioana, A., Peterson, J.: Properly proximal groups and their von Neumann algebras, Preprint arXiv:1809.01881

  6. Boutonnet, R.: \(W^*\)-superrigidity of mixing Gaussian actions of rigid groups. Adv. Math. 244, 69–90 (2013)

    MathSciNet  MATH  Google Scholar 

  7. Burger, M., Mozes, S.: Lattices in products of trees. Inst. Hautes Ètudes Sci. Pub. Sér. I Math. 92, 151–194 (2001)

    MathSciNet  MATH  Google Scholar 

  8. Camm, R.: Simple free products. J. Lond. Math. Soc. 28, 66–76 (1953)

    MathSciNet  MATH  Google Scholar 

  9. Chifan, I., Das, S.: A remark on the ultrapower algebra of the hyperfinite factor. Proc. Amer. Math. Soc. 146, 5289–5294 (2018)

    MathSciNet  MATH  Google Scholar 

  10. Chifan, I., de Santiago, R., Sucpikarnon, W.: Tensor product decompositions of \(\text{II}_1\) factors arising from extensions of amalgamated free product groups , Comm. Math. Phy., 364(218) issue 3, 1163–1194

  11. Chifan, I., Kida, Y.: OE and \(W^{\ast }\) superrigidity results for actions by surface braid groups. Proc. Lond. Math. Soc. 111(6), 1431–1470 (2015)

    MathSciNet  MATH  Google Scholar 

  12. Chifan, I., Ioana, A., Kida, Y.: \(W^*\)-superrigidity for arbitrary actions of central quotients of braid groups. Math. Ann. 361(3–4), 563–582 (2015)

    MathSciNet  MATH  Google Scholar 

  13. Chifan, I., Kida, Y., Pant, S.: Primeness results for von Neumann algebas associated with surface braid groups. Int. Math. Res. Not. 16, 4807–4848 (2016)

    MATH  Google Scholar 

  14. Chifan, I., Peterson, J.: Some unique group measure space decomposition results. Duke Math. J. 162(11), 1923–1966 (2013)

    MathSciNet  MATH  Google Scholar 

  15. Chifan, I., Peterson, J.: On approximation properties for probability measure preserving actions, Preprint (2011)

  16. Chifan, I., Popa, S., Sizemore, O.: Some OE- and \(W^{\ast }\)-rigidity results for actions by wreath product groups. J. Funct. Anal. 263(11), 3422–3448 (2012)

    MathSciNet  MATH  Google Scholar 

  17. Chifan, I., Sinclair, T.: On the structural theory of \(\text{ II}_1\)factors of negatively curved groups. Ann. Sci. Éc. Norm. Sup. 46, 1–33 (2013)

    MATH  Google Scholar 

  18. Chifan, I., Sinclair, T., Udrea, B.: On the structural theory of \(\text{ II}_1\) factors of negatively curved groups, II. Actions by product groups. Adv. Math. 245, 208–236 (2013)

    MathSciNet  MATH  Google Scholar 

  19. Chifan, I., Sinclair, T., Udrea, B.: Inner amenability for groups and central sequences in factors. Ergodic Theory Dyn. Syst. 36, 1106–1129 (2016)

    MathSciNet  MATH  Google Scholar 

  20. Creutz, D., Peterson, J.: Character rigidity for lattices and commensurators, preprint arXiv:1311.4513

  21. Choda, H.: A Galois correspondence in a von Neumann algebra. Tohuku Math. J. 30, 491–504 (1978)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  23. Dahmani, F., Guirardel, V., Osin, D.: (2017) Hyperbolically embedded subgroups and rotating families in groups acting on HYperbolic spaces. Memoirs Amer. Math. Soc. 245 (1156)

  24. Drimbe, D., Hoff, D., Ioana, A.: Prime \(\text{ II}_1\) factors arising from irreducible lattices in products of rank one simple Lie groups, J. Reine. Angew. Math. (to appear), preprint, arXiv:1611.02209

  25. Drimbe, D.: \(W^*\) superrigidity for coinduced actions. Int. J. Math. 29, 1850033 (2018)

    MathSciNet  MATH  Google Scholar 

  26. Evans, D., Kawahigashi, Y.: Quantum symmetries on operator algebras. Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, (1998). xvi+829 pp. ISBN: 0-19-851175-2

  27. Furman, A.: Gromov’s measure equivalence and rigidity of higher rank lattices. Ann. of Math. (2) 150, 1059–1081 (1999)

    MathSciNet  MATH  Google Scholar 

  28. Furstenberg, H.: Ergodic behaviour of diagonal measures and a theorem of Szemeredi on arithmetic progressions. J. Anal. Math. 31, 204–256 (1977)

    MATH  Google Scholar 

  29. Fima, P., Vaes, S.: HNN extensions and unique group measure space decomposition of \(\text{ II}_1\) factors. Trans. Amer. Math. Soc. 354, 2601–2617 (2012)

    MATH  Google Scholar 

  30. Gaboriau, D.: Orbit equivalence and measured group theory, Proceedings of the International Congress of Mathematicians (Hyderabad, India, 2010), Vol. III, 1501-1527, Hindustan Book Agency, New Delhi, (2010)

  31. Ge, L.: On maximal injective subalgebras of factors. Adv. Math. 118, 34–70 (1996)

    MathSciNet  MATH  Google Scholar 

  32. Gaboriau, D., Ioana, A., Tucker-Drob, R.: Cocycle superrigidity for translation actions of product groups, Amer. Journal of Math. (to appear), preprint, arXiv:1603.07616

  33. Ge, L., Kadison, R.: On tensor products of von Neumann algebras. Inventiones Math. 123, 453–466 (1996)

    MathSciNet  MATH  Google Scholar 

  34. Hull, M., Osin, D.: Induced quasi-cocycles on groups with hyperbolically embedded subgroups. Alg. Geom. Topol. 13, 2635–2665 (2013)

    MATH  Google Scholar 

  35. Houdayer, C., Popa, S., Vaes, S.: A class of groups for which every action is \(W^*\)-superrigid. Groups Geom. Dyn. 7, 577–590 (2013)

    MathSciNet  MATH  Google Scholar 

  36. Ioana, A.: Cocycle superrigidity for profinite actions of property (T) groups, ArXiv preprint version, arXiv:0805.2998v1

  37. Ioana, A.: Cocycle superrigidity for profinite actions of property (T) groups. Duke Math J. 157, 337–367 (2011)

    MathSciNet  MATH  Google Scholar 

  38. Ioana, A.: \(W^*\)-superrigidity for Bernoulli actions of property (T) groups. J. Amer. Math. Soc. 24, 1175–1226 (2011)

    MathSciNet  MATH  Google Scholar 

  39. Ioana, A.: Cartan subalgebras of amalgamated free product \(\text{ II}_1\) factors. Ann. Sci. Éc. Norm. Sup. 48, 71–130 (2015)

    Google Scholar 

  40. Ioana, A.: Uniqueness of the group measure space decomposition for Popa’s \(\cal{H}\cal{T}\) factors. Geom. Funct. Anal. 22(3), 699–732 (2012)

    MathSciNet  MATH  Google Scholar 

  41. Ioana, A.: Rigidity for von Neumann algebras, Submitted to Proceedings ICM (2018). Preprint arXiv:1712.00151v1

  42. Ioana, A., Popa, S., Vaes, S.: A class of superrigid group von Neumann algebras. Ann. of Math. (2) 178, 231–286 (2013)

    MathSciNet  MATH  Google Scholar 

  43. Izumi, M., Longo, R., Popa, S.: Automorphisms of von Neumann Algebras with a Generalization to Kac Algebras. J. Functional Anal. 155(1), 25–63 (1998)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  45. Jones, V.F.R.: von Neumann algebras in mathematics and physics, Proceedings of the International Congress of Mathematicians, vol I, II (Kyoto 1990), 121–138, Math Soc. Japan, Tokyo (1991)

  46. Jones, V.F.R.: Subfactor and Knots, CBMS Regional Conference Series in Mathematics 80, AMS (1991)

  47. Jones, V.F.R.: On the origin and development of subfactors and quantum topology. Bull. Amer. Math. Soc. 46, 309–326 (2009)

    MathSciNet  MATH  Google Scholar 

  48. Jiang, Y., Skalski, A.: Maximal subgroups and von Neumann subalgebras with the Haagerup property, (2019). arXiv preprint, arXiv:1903.08190v3

  49. Kida, Y.: Measure equivalence rigidity of the mapping class group. Ann. of Math. (2) 171, 1851–1901 (2010)

    MathSciNet  MATH  Google Scholar 

  50. Krogager, A., Vaes, S.: A class of \(\text{ II}_1\) factors with exactly two group measure space decompositions. J. Math. Pures et Appl. 108, 88–110 (2017)

    MathSciNet  MATH  Google Scholar 

  51. Monod, N., Shalom, Y.: Orbit equivalence rigidity and bounded cohomology. Ann. Math. (2) 164, 825–878 (2006)

    MathSciNet  MATH  Google Scholar 

  52. Neshveyev, S., Størmer, E.: Ergodic theory and maximal abelian subalgebras of the hyperfinite factor. J. Funct. Anal. 195(2), 239–261 (2002)

    MathSciNet  MATH  Google Scholar 

  53. Nielsen, O.A.: Maximal abelian subalgebras of hyperfinite factors II. J. Funct. Anal. 6, 192–202 (1970)

    MathSciNet  MATH  Google Scholar 

  54. Osin, D.: Acylindrically Hyperbolic groups. Trans. Amer. Math. Soc. 368, 851–888 (2016)

    MathSciNet  MATH  Google Scholar 

  55. Ozawa, N., Popa, S.: On a class of \(\text{ II}_1\) factors with at most one Cartan subalgebra. Ann. Math. 172, 713–749 (2010)

    MathSciNet  MATH  Google Scholar 

  56. Ozawa, N., Popa, S.: On a class of \(\text{ II}_1\) factors with at most one Cartan subalgebra II. Am. J. Math. 132(3), 841–866 (2010)

    MATH  Google Scholar 

  57. Packer, J.: Point spectrum of ergodic abelian group actions and the corresponding group-measure space factors. Pacific J. Math 201, 421–428 (2001)

    MathSciNet  Google Scholar 

  58. Peterson, J.: Examples of group actions which are virtually \(W^*\)-superrigid, preprint, arXiv:1002.1745

  59. Peterson, J.: Character rigidity for lattices in higher-rank groups, (2014). preprint, www.math.vanderbilt.edu/~peters10/rigidity.pdf

  60. Peterson, J., Thom, A.: Group cocycles and the ring of affiliated operators. Invent. Math. 185, 561–592 (2011)

    MathSciNet  MATH  Google Scholar 

  61. Pimsner, M., Popa, S.: Entropy and index for subfactors. Ann. Sci. École Norm. Sup. 19, 57–106 (1986)

    MathSciNet  MATH  Google Scholar 

  62. Popa, S.: Notes on Cartan subalgebras in type \(\text{ II}_1\) factors. Math. Scand. 57, 171–188 (1985)

    MathSciNet  Google Scholar 

  63. Popa, S.: Some properties of the symmetric enveloping algebra of a factor, with applications to amenability and property (T), 4 Doc. Math. 665–744, (1999)

  64. Popa, S.: On a class of type \(\text{ II}_1\) factors with Betti numbers invariants. Ann. Math. 163, 809–899 (2006)

    MathSciNet  MATH  Google Scholar 

  65. Popa, S.: Universal construction of subfactors. J. Reine Angew. Math. 543, 39–81 (2002)

    MathSciNet  MATH  Google Scholar 

  66. Popa, S.: Strong rigidity of \(\text{ II}_1\) factors arising from malleable actions of \(w\)-rigid groups I. Invent. Math. 165, 369–408 (2006)

    MathSciNet  MATH  Google Scholar 

  67. Popa, S.: Strong rigidity of \(\text{ II}_1\) factors arising from malleable actions of \(w\)-rigid groups II. Invent. Math. 165, 409–451 (2006)

    MathSciNet  MATH  Google Scholar 

  68. Popa, S.: Deformation and rigidity for group actions and von Neumann algebras, International Congress of Mathematicians. Vol. I, 445–477, Eur. Math. Soc., Zürich, (2007)

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

    MathSciNet  MATH  Google Scholar 

  70. Popa, S.: On the inductive limits of \(\text{ II}_1\) factors with spectral gap. Trans. Amer. Math. Soc. 364, 2987–3000 (2012)

    MathSciNet  MATH  Google Scholar 

  71. Popa, S., Vaes, S.: Group measure space decomposition of \(\text{ II}_1\) factors and \(W^*\)-superrigidity. Invent. Math. 182, 371–417 (2010)

    MathSciNet  MATH  Google Scholar 

  72. Popa, S., Vaes, S.: Unique Cartan decomposition for \(\text{ II}_1\) factors arising from arbitrary actions of free groups. Acta Math. 212, 141–198 (2014)

    MathSciNet  MATH  Google Scholar 

  73. Popa, S., Vaes, S.: Unique Cartan decomposition for \(\text{ II}_1\) factors arising from arbitrary actions of hyperbolic groups. J. Reine Angew. Math. 694, 215–239 (2014)

    MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  75. Suzuki, Y.: Complete descriptions of intermediate operator algebras by intermediate extensions of dynamical systems, To appear in Comm. Math. Phy., ArXiv Preprint, arXiv:1805.02077

  76. Smith, R., White, S., Wiggins, A.: Normalizers of irreducibe subfactors. J. Math. Anal. Appl. 352, 684–695 (2009)

    MathSciNet  MATH  Google Scholar 

  77. Vaes, S.: Rigidity for von Neumann algebras and their invariants, Proceedings of the International Congress of Mathematicians(Hyderabad, India, 2010) Vol III, Hindustan Book Agency, 1624–1650, (2010)

  78. Vaes, S.: One-cohomology and the uniqueness of the group measure space decomposition of a \(\text{ II}_1\) factor. Math. Ann. 355, 661–696 (2013)

    MathSciNet  MATH  Google Scholar 

  79. Zimmer, R.J.: Ergodic actions with generalized discrete spectrum. Illinois J. Math. 20(4), 555–588 (1976)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors are grateful to Adrian Ioana and Jesse Peterson for many helpful discussions related to this project. The authors are extremely grateful to Yuhei Suzuki for carefully reading a first draft of this paper, for his helpful comments and suggestions, and for correcting numerous typos and minor inaccuracies. The authors are also grateful to Rahel Brugger for her helpful comments on our paper, and for correcting a few typos. The second author would like to thank Vaughan Jones for several suggestions and comments regarding the results of this paper. The second author would also like to thank Krishnendu Khan and Pieter Spaas for stimulating conversations regarding the contents of this paper. The first author was partially supported by NSF Grant DMS #1600688. Finally, the authors would like to thank the anonymous referee for many helpful comments and suggestions that greatly improved the exposition of the paper.

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  1. Department of Mathematics, The University of Iowa, 14 MacLean Hall, Iowa City, IA, 52242, USA

    Ionut Chifan & Sayan Das

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  1. Ionut Chifan
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Chifan, I., Das, S. Rigidity results for von Neumann algebras arising from mixing extensions of profinite actions of groups on probability spaces. Math. Ann. 378, 907–950 (2020). https://doi.org/10.1007/s00208-020-02064-8

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