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

### Cyril Houdayer

Université Paris-Sud, Orsay, France### Dimitri L. Shlyakhtenko

University of California Los Angeles, USA### Stefaan Vaes

Katholieke Universiteit Leuven, Belgium

## 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 $\mathbb{R}$ 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.

## Cite this article

Cyril Houdayer, Dimitri L. Shlyakhtenko, Stefaan Vaes, Classification of a family of non-almost-periodic free Araki–Woods factors. J. Eur. Math. Soc. 21 (2019), no. 10, pp. 3113–3142

DOI 10.4171/JEMS/898