# Journal of the European Mathematical Society

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**Volume 21, Issue 10, 2019, pp. 3113–3142**

**DOI: 10.4171/JEMS/898**

Published online: 2019-06-13

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

Cyril Houdayer^{[1]}, Dimitri L. Shlyakhtenko

^{[2]}and Stefaan Vaes

^{[3]}(1) Université Paris-Sud, Orsay, France

(2) University of California Los Angeles, USA

(3) Katholieke Universiteit Leuven, Belgium

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.

*Keywords: *Free Araki–Woods factors, free product von Neumann algebras, Popa’s deformation/ rigidity theory, type III factors

Houdayer Cyril, Shlyakhtenko Dimitri, Vaes Stefaan: Classification of a family of non-almost-periodic free Araki–Woods factors. *J. Eur. Math. Soc.* 21 (2019), 3113-3142. doi: 10.4171/JEMS/898