Structural results for free Araki-Woods factors and their continuous cores

TL;DR

This paper investigates the structural properties of free Araki-Woods factors and their continuous cores, establishing conditions under which these cores are semisolid or solid, and constructing a unique non-amenable solid II_1 factor with full fundamental group.

Contribution

It proves that the continuous core of type III_1 free Araki-Woods factors is semisolid or solid depending on the representation, and constructs a new non-amenable solid II_1 factor with full fundamental group.

Findings

The continuous core is semisolid for any type III_1 free Araki-Woods factor.

If the representation is mixing, the core is solid.

Constructed a non-amenable solid II_1 factor with full fundamental group.

Abstract

We show that for any type ${\rm III_1}$ free Araki-Woods factor $\mathcal{M} = \Gamma(H_\R, U_t)"$ associated with an orthogonal representation $(U_t)$ of $\R$ on a separable real Hilbert space $H_\R$, the continuous core $M = \mathcal{M} \rtimes_\sigma \R$ is a semisolid ${\rm II_\infty}$ factor, i.e. for any non-zero finite projection $q \in M$, the ${\rm II_1}$ factor $qMq$ is semisolid. If the representation $(U_t)$ is moreover assumed to be mixing, then we prove that the core $M$ is solid. As an application, we construct an example of a non-amenable solid ${\rm II_1}$ factor $N$ with full fundamental group, i.e. $\mathcal{F}(N) = \R^*_+$, which is not isomorphic to any interpolated free group factor $L(\F_t)$, for $1 < t \leq +\infty$.