Free Araki-Woods factors and Connes’ bicentralizer problem

Houdayer, Cyril

doi:10.1090/S0002-9939-09-09923-7

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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