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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

A class of $\textrm {II_1}$ 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 $\mathrm {II}_1$ factor $\Gamma (H_{\mathbf {R}})'' \rtimes _\pi \mathbf {Z}$ 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 $\mathrm {II}_1$ factor.


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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

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