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Mathematische Annalen

, Volume 363, Issue 1–2, pp 237–267 | Cite as

Asymptotic structure of free Araki–Woods factors

  • Cyril Houdayer
  • Sven Raum
Article

Abstract

The purpose of this paper is to investigate the structure of Shlyakhtenko’s free Araki–Woods factors using the framework of ultraproduct von Neumann algebras. We first prove that all the free Araki–Woods factors \(\Gamma (H_{\mathbb R}, U_t)^{\prime \prime }\) are \(\omega \)-solid in the following sense: for every von Neumann subalgebra \(Q \subset \Gamma (H_{\mathbb R}, U_t)^{\prime \prime }\) that is the range of a faithful normal conditional expectation and such that the relative commutant \(Q' \cap M^\omega \) is diffuse, we have that \(Q\) is amenable. Next, we prove that the continuous cores of the free Araki–Woods factors \(\Gamma (H_{\mathbb R}, U_t)^{\prime \prime }\) associated with mixing orthogonal representations \(U : \mathbb R \rightarrow \mathcal O(H_{\mathbb R})\) are \(\omega \)-solid type \(\mathrm{II_\infty }\) factors. Finally, when the orthogonal representation \(U : \mathbb R \rightarrow \mathcal O(H_{\mathbb R})\) is weakly mixing, we prove a dichotomy result for all the von Neumann subalgebras \(Q \subset \Gamma (H_{\mathbb R}, U_t)^{\prime \prime }\) that are globally invariant under the modular automorphism group \((\sigma _t^{\varphi _U})\) of the free quasi-free state \(\varphi _U\).

Notes

Acknowledgments

This paper was completed when the first named author was visiting the Research Institute for Mathematical Sciences (RIMS) in Kyoto during Summer 2014. He warmly thanks Narutaka Ozawa and the RIMS for their kind hospitality. The authors also thank Stefaan Vaes for useful remarks regarding a first draft of this manuscript. Finally, the authors thank the anonymous referees for carefully reading the paper and providing valuable comments.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.LAMA UMR 8050CNRS, Université Paris-Est, Marne-la-ValléeMarne-la-Vallée Cedex 2France
  2. 2.RIMSSakyo-kuJapan

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