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Communications in Mathematical Physics

, Volume 336, Issue 2, pp 831–851 | Cite as

Gamma Stability in Free Product von Neumann Algebras

  • Cyril HoudayerEmail author
Article

Abstract

Let \({(M, \varphi) = (M_1, \varphi_1) * (M_2, \varphi_2)}\) be a free product of arbitrary von Neumann algebras endowed with faithful normal states. Assume that the centralizer \({M_1^{\varphi_1}}\) is diffuse. We first show that any intermediate subalgebra \({M_1 \subset Q \subset M}\) which has nontrivial central sequences in M is necessarily equal to M 1. Then we obtain a general structural result for all the intermediate subalgebras \({M_1 \subset Q \subset M}\) with expectation. We deduce that any diffuse amenable von Neumann algebra can be concretely realized as a maximal amenable subalgebra with expectation inside a full nonamenable type III1 factor. This provides the first class of concrete maximal amenable subalgebras in the framework of type III factors. We finally strengthen all these results in the case of tracial free product von Neumann algebras.

Keywords

Free Product Amalgamate Free Product Faithful Normal State Free Group Factor Separable Predual 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. AH12.
    Ando H., Haagerup U.: Ultraproducts of von Neumann algebras. J. Funct. Anal. 266, 6842–6913 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  2. Ba93.
    Barnett L.: Free product von Neumann algebras of type III. Proc. Amer. Math. Soc. 123, 543–553 (1995)zbMATHMathSciNetGoogle Scholar
  3. BC13.
    Boutonnet, R., Carderi, A.: Maximal amenable subalgebras of von Neumann algebras associated with hyperbolic groups. arXiv:1310.5864
  4. Br12.
    Brothier, A.: The cup subalgebra of a II1 factor given by a subfactor planar algebra is maximal amenable. Pacific J. Math. (to appear). arXiv:1210.8091
  5. CFRW08.
    Cameron J., Fang J., Ravichandran M., White S.: The radial masa in a free group factor is maximal injective. J. Lond. Math. Soc. 82, 787–809 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  6. Co72.
    Connes A.: Une classification des facteurs de type III. Ann. Sci. École Norm. Sup. 6, 133–252 (1973)zbMATHMathSciNetGoogle Scholar
  7. Co74.
    Connes A.: Almost periodic states and factors of type III1. J. Funct. Anal. 16, 415–445 (1974)CrossRefzbMATHMathSciNetGoogle Scholar
  8. Co75a.
    Connes A.: Outer conjugacy classes of automorphisms of factors. Ann. Sci. École Norm. Sup. 8, 383–419 (1975)zbMATHMathSciNetGoogle Scholar
  9. Co75b.
    Connes A.: Classification of injective factors. Ann. Math. 104, 73–115 (1976)CrossRefzbMATHMathSciNetGoogle Scholar
  10. Co85.
    Connes, A.: Factors of type III1, property \({{{\rm L}'_\lambda}}\) and closure of inner automorphisms. J. Operator Theory 14, 189–211 (1985)zbMATHMathSciNetGoogle Scholar
  11. Dy92.
    Dykema K.: Factoriality and Connes’ invariant \({T({\mathcal {M}})}\) for free products of von Neumann algebras. J. Reine Angew. Math. 450, 159–180 (1994)zbMATHMathSciNetGoogle Scholar
  12. Fa06.
    Fang J.: On maximal injective subalgebras of tensor products of von Neumann algebras. J. Funct. Anal. 244, 277–288 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  13. Ga09.
    Gao M.: On maximal injective subalgebras. Proc. Am. Math. Soc. 138, 2065–2070 (2010)CrossRefzbMATHGoogle Scholar
  14. Ge95.
    Ge L.: On maximal injective subalgebras of factors. Adv. Math. 118, 34–70 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  15. Ha84.
    Haagerup U.: Connes’ bicentralizer problem and uniqueness of the injective factor of type III1. Acta Math. 158, 95–148 (1987)CrossRefzbMATHMathSciNetGoogle Scholar
  16. Ho07.
    Houdayer C.: Construction of type II1 factors with prescribed countable fundamental group. J. Reine Angew. Math. 634, 169–207 (2009)zbMATHMathSciNetGoogle Scholar
  17. Ho12a.
    Houdayer C.: A class of II1 factors with an exotic Abelian maximal amenable subalgebra. Trans. Am. Math. Soc. 366, 3693–3707 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  18. Ho12b.
    Houdayer C.: Structure of II1 factors arising from free Bogoljubov actions of arbitrary groups. Adv. Math. 260, 414–457 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  19. HS90.
    Haagerup U., Stø rmer E.: Equivalence of normal states on von Neumann algebras and the flow of weights. Adv. Math. 83, 180–262 (1990)CrossRefzbMATHMathSciNetGoogle Scholar
  20. Io12.
    Ioana, A.: Cartan subalgebras of amalgamated free product II1 factors. Ann. Sci. École Norm. Sup. (to appear) arXiv:1207.00541
  21. IPP05.
    Ioana A., Peterson J., Popa S.: Amalgamated free products of w-rigid factors and calculation of their symmetry groups. Acta Math. 200, 85–153 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  22. Jo10.
    Jolissaint, P.: Maximal injective and mixing masas in group factors. arXiv:1004.0128
  23. MvN43.
    Murray F., Neumann J.: Rings of operators. IV1. Ann. Math. 44, 716–808 (1943)CrossRefzbMATHGoogle Scholar
  24. Oc85.
    Ocneanu, A.: Actions of discrete amenable groups on von Neumann algebras. In: Lecture Notes in Mathematics, vol. 1138. Springer, Berlin (1985)Google Scholar
  25. Pe06.
    Peterson J.: L2-rigidity in von Neumann algebras. Invent. Math. 175, 417–433 (2009)CrossRefADSzbMATHMathSciNetGoogle Scholar
  26. Po83.
    Popa S.: Maximal injective subalgebras in factors associated with free groups. Adv. Math. 50, 27–48 (1983)CrossRefzbMATHGoogle Scholar
  27. Po01.
    Popa S.: On a class of type II1 factors with Betti numbers invariants. Ann. Math. 163, 809–899 (2006)CrossRefzbMATHGoogle Scholar
  28. Po03.
    Popa, S.: Strong rigidity of II1 factors arising from malleable actions of w-rigid groups I and II. Invent. Math. 165, 369–408, 409–451 (2006)Google Scholar
  29. Po13.
    Popa S.: A II1 factor approach to the Kadison-Singer problem. Comm. Math. Phys. 332(1), 379–414 (2014)CrossRefADSzbMATHMathSciNetGoogle Scholar
  30. Sh05.
    Shen J.: Maximal injective subalgebras of tensor products of free group factors. J. Funct. Anal. 240, 334–348 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  31. Ta03.
    Takesaki, M.: Theory of operator algebras. II. Encyclopaedia of Mathematical Sciences, vol. 125. Operator Algebras and Non-commutative Geometry, 6. Springer, Berlin (2003)Google Scholar
  32. Ue98.
    Ueda, Y.: Amalgamated free products over Cartan subalgebra. Pacific J. Math. 191, 359–392 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  33. Ue11.
    Ueda Y.: Factoriality, type classification and fullness for free product von Neumann algebras. Adv. Math. 228, 2647–2671 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  34. Vo85.
    Voiculescu, D.-V.: Symmetries of some reduced free product C*-algebras. In: Operator Algebras and Their Connections with Topology and Ergodic Theory. Lecture Notes in Mathematics, vol. 1132, pp. 556–588. Springer, Berlin (1985)Google Scholar
  35. Vo92.
    Voiculescu, D.-V., Dykema, K.J., Nica, A.: Free random variables. In: CRM Monograph Series, vol. 1. American Mathematical Society, Providence (1992)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.CNRS-Université Paris-Est Marne-la-Vallée, LAMA UMR 8050Marne-la-Vallée Cedex 2France

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