Mathematische Annalen

, Volume 363, Issue 1–2, pp 237–267 | Cite as

Asymptotic structure of free Araki–Woods factors

  • Cyril Houdayer
  • Sven Raum


The purpose of this paper is to investigate the structure of Shlyakhtenko’s free Araki–Woods factors using the framework of ultraproduct von Neumann algebras. We first prove that all the free Araki–Woods factors \(\Gamma (H_{\mathbb R}, U_t)^{\prime \prime }\) are \(\omega \)-solid in the following sense: for every von Neumann subalgebra \(Q \subset \Gamma (H_{\mathbb R}, U_t)^{\prime \prime }\) that is the range of a faithful normal conditional expectation and such that the relative commutant \(Q' \cap M^\omega \) is diffuse, we have that \(Q\) is amenable. Next, we prove that the continuous cores of the free Araki–Woods factors \(\Gamma (H_{\mathbb R}, U_t)^{\prime \prime }\) associated with mixing orthogonal representations \(U : \mathbb R \rightarrow \mathcal O(H_{\mathbb R})\) are \(\omega \)-solid type \(\mathrm{II_\infty }\) factors. Finally, when the orthogonal representation \(U : \mathbb R \rightarrow \mathcal O(H_{\mathbb R})\) is weakly mixing, we prove a dichotomy result for all the von Neumann subalgebras \(Q \subset \Gamma (H_{\mathbb R}, U_t)^{\prime \prime }\) that are globally invariant under the modular automorphism group \((\sigma _t^{\varphi _U})\) of the free quasi-free state \(\varphi _U\).



This paper was completed when the first named author was visiting the Research Institute for Mathematical Sciences (RIMS) in Kyoto during Summer 2014. He warmly thanks Narutaka Ozawa and the RIMS for their kind hospitality. The authors also thank Stefaan Vaes for useful remarks regarding a first draft of this manuscript. Finally, the authors thank the anonymous referees for carefully reading the paper and providing valuable comments.


  1. 1.
    Ando, H., Haagerup, U.: Ultraproducts of von Neumann algebras. J. Funct. Anal. 266, 6842–6913 (2014)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Akemann, C.A., Ostrand, P.A.: On a tensor product C\(^{*}\) -algebra associated with the free group on two generators. J. Math. Soc. Japan 27, 589–599 (1975)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Boutonnet, R., Houdayer, C., Raum, S.: Amalgamated free product type III factors with at most one Cartan subalgebra. Compos. Math. 150(1), 143–174 (2014)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Choi, M.D.: A Schwarz inequality for positive linear maps on C\(^{*}\)-algebras. Illinois J. Math. 18, 565–574 (1974)MathSciNetMATHGoogle Scholar
  5. 5.
    Connes, A.: Une classification des facteurs de type III. Ann. Sci. Éc. Norm. Supér. 4(6), 133–252 (1973)MathSciNetGoogle Scholar
  6. 6.
    Connes, A.: Almost periodic states and factors of type III\(_{1}\). J. Funct. Anal. 16, 415–445 (1974)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Connes, A.: Classification of injective factors. Cases II\(_{1}\), II\(_{\infty }\), III\(_{\lambda }, \lambda \ne 1\). Ann. Math. (2) 74, 73–115 (1976)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Houdayer, C.: Sur la classification de certaines algèbres de von Neumann. PhD thesis, Université de Paris VII (2007)Google Scholar
  9. 9.
    Houdayer, C.: Structural results for free Araki-Woods factors and their continuous cores. J. Inst. Math. Jussieu 9(4), 741–767 (2010)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Houdayer, C.: A class of II\(_{1}\) factors with an exotic abelian maximal amenable subalgebra. Trans. Am. Math. Soc. 366, 3693–3707 (2014)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Houdayer, C.: Structure of II\(_{1}\) factors arising from free Bogoljubov actions of arbitrary groups. Adv. Math. 260, 414–457 (2014)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Houdayer, C.: Gamma stability in free product von Neumann algebras. Commun. Math. Phys. (2014). doi: 10.1007/s00220-014-2237-0
  13. 13.
    Houdayer, C., Ricard, É.: Approximation properties and absence of Cartan subalgebra for free Araki-Woods factors. Adv. Math. 228(2), 764–802 (2011)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Ioana, A.: Cartan subalgebras of amalgamated free product II\(_{1}\) factors. Ann. Sci. Éc. Norm. Supér. (to appear, 2015). arXiv:1207.0054
  15. 15.
    Ioana, A., Peterson, J., Popa, S.: Amalgamated free products of weakly rigid factors and calculation of their symmetry groups. Acta Math. 200(1), 85–153 (2008)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Ocneanu, A.: Actions of discrete amenable groups on von Neumann algebras. In: Lecture Notes in Mathematics, vol. 1138. Springer, Berlin (1985)Google Scholar
  17. 17.
    Ozawa, N.: Solid von Neumann algebras. Acta Math. 192(1), 111–117 (2004)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Ozawa, N.: A comment on free group factors. In: Noncommutative harmonic analysis with applications to probability II. Banach Center Publications, vol. 89, pp. 241–245. Polish Acad. Sci. Inst. Math., Warsaw (2010)Google Scholar
  19. 19.
    Peterson, J.: L\(^{2}\)-rigidity in von Neumann algebras. Invent. Math. 175(2), 417–433 (2009)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Popa, S.: Maximal injective subalgebras in factors associated with free groups. Adv. Math. 50, 27–48 (1983)CrossRefMATHGoogle Scholar
  21. 21.
    Popa, S.: Some rigidity results for non-commutative Bernoulli shifts. J. Funct. Anal. 230(2), 273–328 (2006)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Popa, S.: On a class of type II\(_{1}\) factors with Betti numbers invariants. Ann. Math. (2) 163(3), 809–899 (2006)Google Scholar
  23. 23.
    Popa, S.: Strong rigidity of II\(_{1}\) factors arising from malleable actions of \(w\)-rigid groups. I. Invent. Math. 165(2), 369–408 (2006)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Popa, S.: Strong rigidity of II\(_{1}\) factors arising from malleable actions of \(w\)-rigid groups. II. Invent. Math. 165(2), 409–451 (2006)Google Scholar
  25. 25.
    Popa, S.: Deformation and rigidity for group actions and von Neumann algebras. In: Sanz-Solé, M., et al. (eds.) Proceedings of the international congress of mathematicians, Madrid, Spain, August 22–30, 2006. Plenary lectures and ceremonies, vol. I, pp. 445–477. European Mathematical Society, Zürich (2007)Google Scholar
  26. 26.
    Popa, S.: On the superrigidity of malleable actions with spectral gap. J. Am. Math. Soc. 21(4), 981–1000 (2008)CrossRefMATHGoogle Scholar
  27. 27.
    Shlyakhtenko, D.: Free quasi-free states. Pac. J. Math. 177(2), 329–368 (1997)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Shlyakhtenko, D.: Some applications of freeness with amalgamation. J. Reine Angew. Math. 500, 191–212 (1998)MathSciNetMATHGoogle Scholar
  29. 29.
    Shlyakhtenko, D.: \(A\)-valued semicircular systems. J. Funct. Anal. 166(1), 1–47 (1999)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Takesaki, M.: Theory of Operator Algebras III. Springer, Berlin (2003)CrossRefMATHGoogle Scholar
  31. 31.
    Ueda, Y.: On type \({\rm III}_{1}\) factors arising as free products. Math. Res. Lett. 18(5), 909–920 (2011)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Vaes, S.: États quasi-libres libres et facteurs de type \({\rm III}\). In: Séminaire Bourbaki. 2003/2004. Astérisque, vol. 937. Société Mathématique de France, Paris (2007)Google Scholar
  33. 33.
    Vaes, S., Vergnioux, R.: The boundary of universal discrete quantum groups, exactness, and factoriality. Duke Math. J. 140(1), 35–84 (2007)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

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

Personalised recommendations