Skip to main content
SpringerLink
Log in

Gamma Stability in Free Product von Neumann Algebras

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ando H., Haagerup U.: Ultraproducts of von Neumann algebras. J. Funct. Anal. 266, 6842–6913 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  2. Barnett L.: Free product von Neumann algebras of type III. Proc. Amer. Math. Soc. 123, 543–553 (1995)

    MATH  MathSciNet  Google Scholar 

  3. Boutonnet, R., Carderi, A.: Maximal amenable subalgebras of von Neumann algebras associated with hyperbolic groups. arXiv:1310.5864

  4. 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. 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)

    Article  MATH  MathSciNet  Google Scholar 

  6. Connes A.: Une classification des facteurs de type III. Ann. Sci. École Norm. Sup. 6, 133–252 (1973)

    MATH  MathSciNet  Google Scholar 

  7. Connes A.: Almost periodic states and factors of type III1. J. Funct. Anal. 16, 415–445 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  8. Connes A.: Outer conjugacy classes of automorphisms of factors. Ann. Sci. École Norm. Sup. 8, 383–419 (1975)

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  10. Connes, A.: Factors of type III1, property \({{{\rm L}'_\lambda}}\) and closure of inner automorphisms. J. Operator Theory 14, 189–211 (1985)

    MATH  MathSciNet  Google Scholar 

  11. Dykema K.: Factoriality and Connes’ invariant \({T({\mathcal {M}})}\) for free products of von Neumann algebras. J. Reine Angew. Math. 450, 159–180 (1994)

    MATH  MathSciNet  Google Scholar 

  12. Fang J.: On maximal injective subalgebras of tensor products of von Neumann algebras. J. Funct. Anal. 244, 277–288 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  13. Gao M.: On maximal injective subalgebras. Proc. Am. Math. Soc. 138, 2065–2070 (2010)

    Article  MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  15. Haagerup U.: Connes’ bicentralizer problem and uniqueness of the injective factor of type III1. Acta Math. 158, 95–148 (1987)

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  17. Houdayer C.: A class of II1 factors with an exotic Abelian maximal amenable subalgebra. Trans. Am. Math. Soc. 366, 3693–3707 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  18. Houdayer C.: Structure of II1 factors arising from free Bogoljubov actions of arbitrary groups. Adv. Math. 260, 414–457 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  19. Haagerup U., Stø rmer E.: Equivalence of normal states on von Neumann algebras and the flow of weights. Adv. Math. 83, 180–262 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  20. Ioana, A.: Cartan subalgebras of amalgamated free product II1 factors. Ann. Sci. École Norm. Sup. (to appear) arXiv:1207.00541

  21. 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)

    Article  MATH  MathSciNet  Google Scholar 

  22. Jolissaint, P.: Maximal injective and mixing masas in group factors. arXiv:1004.0128

  23. Murray F., Neumann J.: Rings of operators. IV1. Ann. Math. 44, 716–808 (1943)

    Article  MATH  Google Scholar 

  24. Ocneanu, A.: Actions of discrete amenable groups on von Neumann algebras. In: Lecture Notes in Mathematics, vol. 1138. Springer, Berlin (1985)

  25. Peterson J.: L2-rigidity in von Neumann algebras. Invent. Math. 175, 417–433 (2009)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  26. Popa S.: Maximal injective subalgebras in factors associated with free groups. Adv. Math. 50, 27–48 (1983)

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  28. 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)

  29. Popa S.: A II1 factor approach to the Kadison-Singer problem. Comm. Math. Phys. 332(1), 379–414 (2014)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  30. Shen J.: Maximal injective subalgebras of tensor products of free group factors. J. Funct. Anal. 240, 334–348 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  31. Takesaki, M.: Theory of operator algebras. II. Encyclopaedia of Mathematical Sciences, vol. 125. Operator Algebras and Non-commutative Geometry, 6. Springer, Berlin (2003)

  32. Ueda, Y.: Amalgamated free products over Cartan subalgebra. Pacific J. Math. 191, 359–392 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  33. Ueda Y.: Factoriality, type classification and fullness for free product von Neumann algebras. Adv. Math. 228, 2647–2671 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  34. 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)

  35. Voiculescu, D.-V., Dykema, K.J., Nica, A.: Free random variables. In: CRM Monograph Series, vol. 1. American Mathematical Society, Providence (1992)

Download references

Author information

Authors and Affiliations

  1. CNRS-Université Paris-Est Marne-la-Vallée, LAMA UMR 8050, 77454, Marne-la-Vallée Cedex 2, France

    Cyril Houdayer

Authors
  1. Cyril Houdayer
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Cyril Houdayer.

Additional information

Communicated by Y. Kawahigashi

Research supported by ANR grant NEUMANN.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Houdayer, C. Gamma Stability in Free Product von Neumann Algebras. Commun. Math. Phys. 336, 831–851 (2015). https://doi.org/10.1007/s00220-014-2237-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-014-2237-0

Keywords

Search

Navigation