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

Abstract

We show that for any type $\mathrm{II}{\mathrm{I}}_1$ free Araki-Woods factor $\mathcal{M}=\Gamma (H_{\mathbb{R}},U_t)"$ associated with an orthogonal representation $(U_t)$ of $\mathbb{R}$ on a separable real Hilbert space $H_{\mathbb{R}}$, the continuous core $M=\mathcal{M}{⋊}_{\sigma }\mathbb{R}$ is a semisolid $\mathrm{I}{\mathrm{I}}_{\infty }$ factor, i.e. for any non-zero finite projection $q\in M$, the $\mathrm{I}{\mathrm{I}}_1$ 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 $\mathrm{I}{\mathrm{I}}_1$ factor $N$ with full fundamental group, i.e. $\mathcal{F}(N)={\mathbb{R}}_{+}^{\ast }$, which is not isomorphic to any interpolated free group factor $L(\F_t)$, for $1<t\le +\infty$.

22 pages

References in corpus (5)

• On a class of ${\mathrm{II}}_1$ factors with at most one Cartan subalgebra
• Construction of type $\mathrm{I}{\mathrm{I}}_1$ factors with prescribed countable fundamental group
• Free Araki-Woods factors and Connes' bicentralizer problem
• On some free products of von Neumann algebras which are free Araki-Woods factors
• On multiplicity and free absorption for free Araki-Woods factors

Cited by in corpus (8)

• Approximation properties and absence of Cartan subalgebra for free Araki-Woods factors
• Strongly solid $\mathrm{I}{\mathrm{I}}_1$ factors with an exotic MASA
• Asymptotic structure of free Araki-Woods factors
• Strong solidity of free Araki-Woods factors
• Classification of a family of non almost periodic free Araki-Woods factors
• Examples of factors which have no Cartan subalgebras
• A characterization of fullness of continuous cores of type III${}_1$ free product factors
• Absence of Cartan subalgebras in continuous cores of free product von Neumann algebras