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Asymptotic structure of free Araki-Woods factors. (English) Zbl 1339.46057
Summary: 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 \(\Gamma (H_{\mathbb R}, U_t)^{''}\) are \(\omega \)-solid in the following sense: for every von Neumann subalgebra \(Q \subset \Gamma (H_{\mathbb R}, U_t)^{''}\) 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 \(\Gamma (H_{\mathbb R}, U_t)^{''}\) associated with mixing orthogonal representations \(U : \mathbb R \to \mathcal O(H_{\mathbb R})\) are \(\omega \)-solid type \(\mathrm{II}_\infty \) factors. Finally, when the orthogonal representation \(U : \mathbb R \to \mathcal O(H_{\mathbb R})\) is weakly mixing, we prove a dichotomy result for all the von Neumann subalgebras \(Q \subset \Gamma (H_{\mathbb R}, U_t)^{''}\) that are globally invariant under the modular automorphism group \((\sigma _t^{\varphi _U})\) of the free quasi-free state \(\varphi _U\).

46L36 Classification of factors
46L10 General theory of von Neumann algebras
46L40 Automorphisms of selfadjoint operator algebras
Full Text: DOI arXiv
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