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A class of $ {II_1}$ factors with an exotic abelian maximal amenable subalgebra


Author: Cyril Houdayer
Journal: Trans. Amer. Math. Soc. 366 (2014), 3693-3707
MSC (2010): Primary 46L10, 46L54, 46L55, 22D25
DOI: https://doi.org/10.1090/S0002-9947-2014-05964-3
Published electronically: March 20, 2014
MathSciNet review: 3192613
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Abstract: We show that for every mixing orthogonal representation $ \pi : \mathbf {Z} \to \mathcal O(H_{\mathbf {R}})$, the abelian subalgebra $ \mathrm {L}(\mathbf {Z})$ is maximal amenable in the crossed product $ {\rm II}_1$ factor $ \Gamma (H_{\mathbf {R}})^{\prime \prime } \rtimes _\pi \mathbf {Z}$ associated with the free Bogoljubov action of the representation $ \pi $. This provides uncountably many non-isomorphic $ A$-$ A$-bimodules which are disjoint from the coarse $ A$-$ A$-bimodule and of the form $ \mathrm {L}^2(M \ominus A)$ where $ A \subset M$ is a maximal amenable masa in a $ {\rm II_1}$ factor.


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  • [1] A. Brothier, The cup subalgebra of a $ II_1$ factor given by a subfactor planar algebra is maximal amenable. arXiv:1210.8091
  • [2] Nathanial P. Brown and Narutaka Ozawa, $ C^*$-algebras and finite-dimensional approximations, Graduate Studies in Mathematics, vol. 88, American Mathematical Society, Providence, RI, 2008. MR 2391387 (2009h:46101)
  • [3] Jan Cameron, Junsheng Fang, Mohan Ravichandran, and Stuart White, The radial masa in a free group factor is maximal injective, J. Lond. Math. Soc. (2) 82 (2010), no. 3, 787-809. MR 2739068 (2012d:46135), https://doi.org/10.1112/jlms/jdq052
  • [4] A. Connes, Classification of injective factors. Cases $ II_{1},$ $ II_{\infty },$ $ III_{\lambda },$ $ \lambda \not =1$, Ann. of Math. (2) 104 (1976), no. 1, 73-115. MR 0454659 (56 #12908)
  • [5] I. P. Cornfeld, S. V. Fomin, and Ya. G. Sinaĭ, Ergodic theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 245, Springer-Verlag, New York, 1982. Translated from the Russian by A. B. Sosinskiĭ. MR 832433 (87f:28019)
  • [6] Ken Dykema, Interpolated free group factors, Pacific J. Math. 163 (1994), no. 1, 123-135. MR 1256179 (95c:46103)
  • [7] K. Dykema and K. Mukherjee, Measure-multiplicity of the Laplacian masa. Glasgow Math. J. 55 (2013), no. 2, 285-292. MR 3040862
  • [8] Cyril Houdayer and Éric Ricard, Approximation properties and absence of Cartan subalgebra for free Araki-Woods factors, Adv. Math. 228 (2011), no. 2, 764-802. MR 2822210 (2012f:46116), https://doi.org/10.1016/j.aim.2011.06.010
  • [9] Cyril Houdayer and Dimitri Shlyakhtenko, Strongly solid $ {\rm II}_1$ factors with an exotic MASA, Int. Math. Res. Not. IMRN 6 (2011), 1352-1380. MR 2806507 (2012e:46133)
  • [10] Richard V. Kadison, Diagonalizing matrices, Amer. J. Math. 106 (1984), no. 6, 1451-1468. MR 765586 (86d:46056), https://doi.org/10.2307/2374400
  • [11] Jean-Pierre Kahane and Raphaël Salem, Ensembles parfaits et séries trigonométriques, 2nd ed., Hermann, Paris, 1994 (French, with French summary). With notes by Kahane, Thomas W. Körner, Russell Lyons and Stephen William Drury. MR 1303593 (96e:42001)
  • [12] Alexander S. Kechris, Global aspects of ergodic group actions, Mathematical Surveys and Monographs, vol. 160, American Mathematical Society, Providence, RI, 2010. MR 2583950 (2011b:37003)
  • [13] Alexander S. Kechris and Alain Louveau, Descriptive set theory and the structure of sets of uniqueness, London Mathematical Society Lecture Note Series, vol. 128, Cambridge University Press, Cambridge, 1987. MR 953784 (90a:42008)
  • [14] F. J. Murray and J. von Neumann, On rings of operators. IV, Ann. of Math. (2) 44 (1943), 716-808. MR 0009096 (5,101a)
  • [15] Sergey Neshveyev and Erling Størmer, Ergodic theory and maximal abelian subalgebras of the hyperfinite factor, J. Funct. Anal. 195 (2002), no. 2, 239-261. MR 1940356 (2003j:46106), https://doi.org/10.1006/jfan.2002.3967
  • [16] Narutaka Ozawa, Solid von Neumann algebras, Acta Math. 192 (2004), no. 1, 111-117. MR 2079600 (2005e:46115), https://doi.org/10.1007/BF02441087
  • [17] Narutaka Ozawa and Sorin Popa, On a class of $ {\rm II}_1$ factors with at most one Cartan subalgebra, Ann. of Math. (2) 172 (2010), no. 1, 713-749. MR 2680430 (2011j:46101), https://doi.org/10.4007/annals.2010.172.713
  • [18] Sorin Popa, Maximal injective subalgebras in factors associated with free groups, Adv. in Math. 50 (1983), no. 1, 27-48. MR 720738 (85h:46084), https://doi.org/10.1016/0001-8708(83)90033-6
  • [19] Sorin Popa, Strong rigidity of $ \rm II_1$ factors arising from malleable actions of $ w$-rigid groups. I, Invent. Math. 165 (2006), no. 2, 369-408. MR 2231961 (2007f:46058), https://doi.org/10.1007/s00222-006-0501-4
  • [20] Florin Rădulescu, Singularity of the radial subalgebra of $ {\mathcal {L}}(F_N)$ and the Pukánszky invariant, Pacific J. Math. 151 (1991), no. 2, 297-306. MR 1132391 (93b:46120)
  • [21] Florin Rădulescu, Random matrices, amalgamated free products and subfactors of the von Neumann algebra of a free group, of noninteger index, Invent. Math. 115 (1994), no. 2, 347-389. MR 1258909 (95c:46102), https://doi.org/10.1007/BF01231764
  • [22] Walter Rudin, Fourier analysis on groups, Interscience Tracts in Pure and Applied Mathematics, No. 12, Interscience Publishers (a division of John Wiley and Sons), New York-London, 1962. MR 0152834 (27 #2808)
  • [23] D.-V. Voiculescu, 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. MR 799593 (87d:46075)
  • [24] D. V. Voiculescu, K. J. Dykema, and A. Nica, Free random variables, CRM Monograph Series, vol. 1, American Mathematical Society, Providence, RI, 1992. A noncommutative probability approach to free products with applications to random matrices, operator algebras and harmonic analysis on free groups. MR 1217253 (94c:46133)
  • [25] D. Voiculescu, The analogues of entropy and of Fisher's information measure in free probability theory. III. The absence of Cartan subalgebras, Geom. Funct. Anal. 6 (1996), no. 1, 172-199. MR 1371236 (96m:46119), https://doi.org/10.1007/BF02246772

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Additional Information

Cyril Houdayer
Affiliation: Unité de Mathématiques Pures et Appliquées, École Normale Supérieure de Lyon, CNRS-UMR 5669, 69364 Lyon Cedex 7, France
Email: cyril.houdayer@ens-lyon.fr

DOI: https://doi.org/10.1090/S0002-9947-2014-05964-3
Keywords: Free Gaussian functor, maximal amenable subalgebras, asymptotic orthogonality property, Rajchman measures
Received by editor(s): April 30, 2012
Received by editor(s) in revised form: September 21, 2012
Published electronically: March 20, 2014
Additional Notes: The author’s research was partially supported by ANR grants AGORA NT09-461407 and NEUMANN
Article copyright: © Copyright 2014 American Mathematical Society

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