Asymptotic structure of free Araki-Woods factors

TL;DR

This paper studies the asymptotic structure of free Araki-Woods factors, proving their $ ext{ω}$-solidity and analyzing the properties of their continuous cores, with results depending on the nature of the underlying orthogonal representation.

Contribution

It establishes $ ext{ω}$-solidity of free Araki-Woods factors and their continuous cores, providing new structural insights based on the properties of the orthogonal representation.

Findings

All free Araki-Woods factors are ω-solid.

Continuous cores of these factors are ω-solid type II∞ factors.

A dichotomy is proved for invariant subalgebras under the modular automorphism group.

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 $U : \mathbb R \to \mathcal O(H_{\mathbb R})$ are $\omega$-solid type ${\rm II_\infty}$ 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 $Q \subset \Gamma(H_{\mathbb R}, U_t)^{\prime \prime}$ that are globally invariant under the modular automorphism group $(\sigma_t^{\varphi_U})$ of the free quasi-free state $\varphi_U$.