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

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 $\mathcal{C}({\bigvee }_{k\ge 1}{\mu }^{\ast k})$ 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.

v2: minor changes, final version, to appear in Journal of the European Mathematical Society

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• Pointwise inner automorphisms of almost periodic factors
• Cocycle Conjugacy of Free Bogoljubov Actions of $\mathbb{R}$
• Classification results for nonsingular Bernoulli crossed products
• Properties of solutions of the "naive" functional Schroedinger equation for QCD