Classification of a family of non almost periodic free Araki-Woods factors

TL;DR

This paper classifies a broad class of non almost periodic free Araki-Woods factors using measure invariants, providing new rigidity results and a deformation/rigidity criterion for von Neumann algebra isomorphisms.

Contribution

It introduces a complete classification method for certain free Araki-Woods factors based on measure class invariants and establishes a deformation/rigidity criterion for unitary conjugacy.

Findings

Classification of free Araki-Woods factors via measure invariants

Deformation/rigidity criterion for faithful normal states

Rigidity results for free product von Neumann algebras

Abstract

We obtain a complete classification of a large class of non almost periodic free Araki-Woods factors $\Gamma(\mu,m)"$ up to isomorphism. We do this by showing that free Araki-Woods factors $\Gamma(\mu, m)"$ arising from finite symmetric Borel measures $\mu$ on $\mathbf{R}$ whose atomic part $\mu_a$ is nonzero and not concentrated on $\{0\}$ have the joint measure class $\mathcal C(\bigvee_{k \geq 1} \mu^{\ast k})$ as an invariant. Our key technical result is a deformation/rigidity criterion for the unitary conjugacy of two faithful normal states. We use this to also deduce rigidity and classification theorems for free product von Neumann algebras.