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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 $$\varphi _U$$.

##### MSC:
 46L36 Classification of factors 46L10 General theory of von Neumann algebras 46L40 Automorphisms of selfadjoint operator algebras
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