Asymptotic structure of free Araki-Woods factors

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 $\mathrm{\Gamma }(H_{\mathbb{R}},U_t{)}^{\mathrm{\prime }\mathrm{\prime }}$ are $\omega$-solid in the following sense: for every von Neumann subalgebra $Q\subset \mathrm{\Gamma }(H_{\mathbb{R}},U_t{)}^{\mathrm{\prime }\mathrm{\prime }}$ that is the range of a faithful normal conditional expectation and such that the relative commutant $Q^{\mathrm{\prime }}\cap M^{\omega }$ is diffuse, we have that $Q$ is amenable. Next, we prove that the continuous cores of the free Araki-Woods factors $\mathrm{\Gamma }(H_{\mathbb{R}},U_t{)}^{\mathrm{\prime }\mathrm{\prime }}$ associated with mixing orthogonal representations $U:\mathbb{R}\to \mathcal{O}(H_{\mathbb{R}})$ are $\omega$-solid type $\mathrm{I}{\mathrm{I}}_{\mathrm{\infty }}$ 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 $Q\subset \mathrm{\Gamma }(H_{\mathbb{R}},U_t{)}^{\mathrm{\prime }\mathrm{\prime }}$ that are globally invariant under the modular automorphism group $({\sigma }_t^{{\phi }_U})$ of the free quasi-free state ${\phi }_U$.