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Asymptotic structure of free product von Neumann algebras

Published online by Cambridge University Press:  20 May 2016

CYRIL HOUDAYER  and
YOSHIMICHI UEDA
CYRIL HOUDAYER
Affiliation:
Laboratoire de Mathématiques d'Orsay, Université Paris-Sud, CNRS, Université Paris-Saclay, 91405 Orsay, FRANCE. e-mails: [email protected]
YOSHIMICHI UEDA
Affiliation:
Graduate School of Mathematics, Kyushu University, Fukuoka, 810-8560, JAPAN. e-mails: [email protected]
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Abstract

Let (M, ϕ) = (M 1, ϕ1) * (M 2, ϕ2) be the free product of any σ-finite von Neumann algebras endowed with any faithful normal states. We show that whenever Q ⊂ M is a von Neumann subalgebra with separable predual such that both Q and Q ∩ M 1 are the ranges of faithful normal conditional expectations and such that both the intersection Q ∩ M 1 and the central sequence algebra Q′ ∩ Mω are diffuse (e.g. Q is amenable), then Q must sit inside M 1. This result generalizes the previous results of the first named author in [Ho14] and moreover completely settles the questions of maximal amenability and maximal property Gamma of the inclusion M 1 ⊂ M in arbitrary free product von Neumann algebras.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 161 , Issue 3 , November 2016 , pp. 489 - 516
Copyright
Copyright © Cambridge Philosophical Society 2016 

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References

REFERENCES

[AH12] Ando, H. and Haagerup, U. Ultraproducts of von Neumann algebras. J. Funct. Anal. 266 (2014), 6842–6913.CrossRefGoogle Scholar
[BC14] Boutonnet, R. and Carderi, A. Maximal amenable von Neumann subalgebras arising from maximal amenable subgroups. Geom. Funct. Anal. 25 (2015), 1688–1705.CrossRefGoogle Scholar
[BHR12] Boutonnet, R., Houdayer, C. and Raum, S. Amalgamated free product type III factors with at most one Cartan subalgebra. Compositio Math. 150 (2014), 143–174.CrossRefGoogle Scholar
[Bl06] Blackadar, B. Operator Algebras. Encyclopaedia of Mathematical Sciences, 122. Operator Algebras and Non-commutative Geometry 3. (Springer-Verlag, Berlin, 2006), xx+517 pp.CrossRefGoogle Scholar
[CH08] Chifan, I. and Houdayer, C. Bass–Serre rigidity results in von Neumann algebras. Duke Math. J. 153 (2010), 23–54.CrossRefGoogle Scholar
[Co72] Connes, A. Une classification des facteurs de type III. Ann. Sci. École Norm. Sup. 6 (1973), 133–252.CrossRefGoogle Scholar
[Co75] Connes, A. Classification of injective factors. Cases II1, II∞, IIIλ, λ 1. Ann. of Math. 74 (1976), 73–115.CrossRefGoogle Scholar
[Co76] Connes, A. On the cohomology of operator algebras. J. Funct. Anal. 28 (1978), 248–253.CrossRefGoogle Scholar
[CS78] Connes, A. and Størmer, E. Homogeneity of the state space of factors of type III1 . J. Funct. Anal. 28 (1978), 187–196.CrossRefGoogle Scholar
[FM75] Feldman, J. and Moore, C.C. Ergodic equivalence relations, cohomology and von Neumann algebras. I and II. Trans. Amer. Math. Soc. 234 (1977), 289–324, 325–359.CrossRefGoogle Scholar
[Ha85] Haagerup, U. Connes' bicentralizer problem and uniqueness of the injective factor of type III1 . Acta Math. 69 (1986), 95–148.Google Scholar
[HS90] Haagerup, U. and Størmer, E. Equivalence of normal states on von Neumann algebras and the flow of weights. Adv. Math. 83 (1990), 180–262.CrossRefGoogle Scholar
[Ho12a] Houdayer, C. A class of II1 factors with an exotic abelian maximal amenable subalgebra. Trans. Amer. Math. Soc. 366 (2014), 3693–3707.CrossRefGoogle Scholar
[Ho12b] Houdayer, C. Structure of II1 factors arising from free Bogoljubov actions of arbitrary groups. Adv. Math. 260 (2014), 414–457.CrossRefGoogle Scholar
[Ho14] Houdayer, C. Gamma stability in free product von Neumann algebras. Commun. Math. Phys. 336 (2015), 831–851.CrossRefGoogle Scholar
[HI15] Houdayer, C. and Isono, Y. Unique prime factorization and bicentralizer problem for a class of type III factors. arXiv:1503.01388 Google Scholar
[HR14] Houdayer, C. and Raum, S. Asymptotic structure of free Araki–Woods factors. Math. Ann. 363 (2015), 237–267.CrossRefGoogle Scholar
[HR10] Houdayer, C. and Ricard, É.. Approximation properties and absence of Cartan subalgebra for free Araki–Woods factors. Adv. Math. 228 (2011), 764–802.CrossRefGoogle Scholar
[HU15] Houdayer, C. and Ueda, Y.,. Rigidity of free product von Neumann algebra. To appear in Compositio Math. arXiv:1507.02157 Google Scholar
[HV12] Houdayer, C. and Vaes, S. Type III factors with unique Cartan decomposition. J. Math. Pure Appl. 100 (2013), 564–590.CrossRefGoogle Scholar
[Io12] Ioana, A. Cartan subalgebras of amalgamated free product II1 factors. Ann. Sci. École Norm. Sup. 48 (2015), 71–130.CrossRefGoogle Scholar
[IPP05] Ioana, A., Peterson, J. and Popa, S. Amalgamated free products of w-rigid factors and calculation of their symmetry groups. Acta Math. 200 (2008), 85–153.CrossRefGoogle Scholar
[Jo82] Jones, V.F.R. Index for subfactors. Invent. Math. 72 (1983), 1–25.CrossRefGoogle Scholar
[Ka82] Kadison, R.V. Diagonalizing matrices. Amer. J. Math. 106 (1984), 1451–1468.CrossRefGoogle Scholar
[Ko88] Kosaki, H. Characterization of crossed product (properly infinite case). Pacific J. Math. 137 (1989), 159–167.CrossRefGoogle Scholar
[Kr75] Krieger, W. On ergodic flows and the isomorphism of factors. Math. Ann. 223 (1976),19–70.CrossRefGoogle Scholar
[MU12] Martín, M. and Ueda, Y. On the geometry of von Neumann algebra preduals. Positivity 18 (2014), 519–530.CrossRefGoogle Scholar
[MT13] Masuda, T. and Tomatsu, R. Classification of actions of discrete Kac algebras on injective factors. To appear in Mem. Amer. Math. Soc. arXiv:1306.5046 Google Scholar
[Oc85] Ocneanu, A. Actions of discrete amenable groups on von Neumann algebras. Lecture Notes in Mathematics, 1138 (Springer-Verlag, Berlin, 1985), iv+115 pp.CrossRefGoogle Scholar
[Oz15] Ozawa, N. A remark on amenable von Neumann subalgebras in a tracial free product. Proc. Japan Acad. Ser. A Math. Sci. 91 (2015), 104.CrossRefGoogle Scholar
[Pe06] Peterson, J. L2-rigidity in von Neumann algebras. Invent. Math. 175 (2009), 417–433.CrossRefGoogle Scholar
[PP84] Pimsner, M. and Popa, S. Entropy and index for subfactors. Ann. Sci. École Norm. Sup. 19 (1986), 57–106.CrossRefGoogle Scholar
[Po83] Popa, S. Maximal injective subalgebras in factors associated with free groups. Adv. Math. 50 (1983), 27–48.CrossRefGoogle Scholar
[Po90] Popa, S. Markov traces on universal Jones algebras and subfactors of finite index. Invent. Math. 111 (1993), 375–405.CrossRefGoogle Scholar
[Po01] Popa, S. On a class of type II1 factors with Betti numbers invariants. Ann. of Math. 163 (2006), 809–899.CrossRefGoogle Scholar
[Po03] Popa, S. Strong rigidity of II1 factors arising from malleable actions of w-rigid groups I. Invent. Math. 165 (2006), 369–408.CrossRefGoogle Scholar
[Po06] Popa, S. On the superrigidity of malleable actions with spectral gap. J. Amer. Math. Soc. 21 (2008), 981–1000.CrossRefGoogle Scholar
[Ta02] Takesaki, M. Theory of Operator Algebras. I. Encyclopedia of Mathematical Sciences, 124. Operator Algebras and Non-commutative Geometry, 5 (Springer, Berlin, 2002), xx+415 pp.Google Scholar
[Ta03] Takesaki, M. Theory of operator algebras. II. Encyclopaedia of Mathematical Sciences, 125. Operator Algebras and Non-commutative Geometry, 6 (Springer-Verlag, Berlin, 2003), xxii+518 pp.CrossRefGoogle Scholar
[Ue98a] Ueda, Y. Amalgamated free products over Cartan subalgebra. Pacific J. Math. 191 (1999), 359–392.CrossRefGoogle Scholar
[Ue98b] Ueda, Y. Remarks on free products with respect to non-tracial states. Math. Scand. 88 (2001), 111–125.CrossRefGoogle Scholar
[Ue00] Ueda, Y. Fullness, Connes' χ-groups, and ultra-products of amalgamated free products over Cartan subalgebras. Trans. Amer. Math. Soc. 355 (2003), 349–371.CrossRefGoogle Scholar
[Ue10] Ueda, Y. Factoriality, type classification and fullness for free product von Neumann algebras. Adv. Math. 228 (2011), 2647–2671.CrossRefGoogle Scholar
[Ue11] Ueda, Y. On type III1 factors arising as free products. Math. Res. Lett. 18 (2011), 909–920.CrossRefGoogle Scholar
[Ue12] Ueda, Y. Some analysis on amalgamated free products of von Neumann algebras in non-tracial setting. J. London Math. Soc. 88 (2013), 25–48.CrossRefGoogle Scholar
[Va06] Vaes, S. Rigidity results for Bernoulli actions and their von Neumann algebras (after Sorin Popa). Astérisque 311 (2007), 237–294.Google Scholar
[Va07] Vaes, S. Explicit computations of all finite index bimodules for a family of II1 factors. Ann. Sci. École Norm. Sup. 41 (2008), 743–788.CrossRefGoogle Scholar
[Vo85] Voiculescu, D.-V.. Symmetries of some reduced free product C*-algebras. Operator algebras and their Connections with Topology and Ergodic Theory. Lecture Notes in Mathematics 1132 (Springer-Verlag, 1985), 556–588.Google Scholar
[VDN92] Voiculescu, D.-V., Dykema, K.J. and Nica, A.. Free random variables. CRM Monograph Series 1 (American Mathematical Society, Providence, RI, 1992).CrossRefGoogle Scholar