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 $\Gamma (\mu ,m{)}^{\prime \prime }$ up to isomorphism. We do this by showing that free Araki–Woods factors $\Gamma (\mu ,m{)}^{\prime \prime }$ 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 $\mathcal{C}({\bigvee }_{k\ge 1}{\mu }^{\ast k})$ 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.