Asymptotic structure of free Araki–Woods factors

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 $\varphi _U$.