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 $Q^{\prime }\cap M^{\omega }$ is diffuse, we have that $Q$ is amenable. Next, we prove that the continuous cores of the free Araki-Woods factors $\Gamma (H_{\mathbb{R}},U_t{)}^{\prime \prime }$ associated with mixing orthogonal representations $U:\mathbb{R}\to \mathcal{O}(H_{\mathbb{R}})$ are $\omega$-solid type $\mathrm{I}{\mathrm{I}}_{\infty }$ 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 $Q\subset \Gamma (H_{\mathbb{R}},U_t{)}^{\prime \prime }$ that are globally invariant under the modular automorphism group $({\sigma }_t^{{\phi }_U})$ of the free quasi-free state ${\phi }_U$.

29 pages. v2: final version, to appear in Math. Ann

References in corpus (1)

• Gamma stability in free product von Neumann algebras

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