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Proceedings of the American Mathematical Society

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Free Araki-Woods factors and Connes’ bicentralizer problem


Author: Cyril Houdayer
Journal: Proc. Amer. Math. Soc. 137 (2009), 3749-3755
MSC (2000): Primary 46L10, 46L54
DOI: https://doi.org/10.1090/S0002-9939-09-09923-7
Published electronically: May 21, 2009
MathSciNet review: 2529883
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Abstract: We show that for any type $\mathrm {III}_1$ free Araki-Woods factor $\mathcal {M} = \Gamma (H_{\mathbf {R}}, U_t)''$, the bicentralizer of the free quasi-free state $\varphi _U$ is trivial. Using Haagerup’s Theorem, it follows that there always exists a faithful normal state $\psi$ on $\mathcal {M}$ such that $(\mathcal {M}^\psi )’ \cap \mathcal {M} = \mathbf {C}$.


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Additional Information

Cyril Houdayer
Affiliation: Department of Mathematics, University of California, Los Angeles, 520 Portola Plaza, Los Angeles, California 90095
Address at time of publication: CNRS-ENS Lyon, UMPA UMR 5669, 69364 Lyon cedex 7, France
Email: cyril@math.ucla.edu, cyril.houdayer@umpa.ens-lyon.fr

Keywords: Free Araki-Woods factors, Connes’ bicentralizer problem
Received by editor(s): October 7, 2008
Received by editor(s) in revised form: February 16, 2009
Published electronically: May 21, 2009
Communicated by: Marius Junge
Article copyright: © Copyright 2009 American Mathematical Society