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Classification of a family of non-almost-periodic free Araki-Woods factors. (English) Zbl 1434.46037
Summary: 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 $$\mathcal C(\bigvee_{k \geq 1} \mu^{\ast k})$$ 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.

##### MSC:
 46L36 Classification of factors 46L10 General theory of von Neumann algebras 46L09 Free products of $$C^*$$-algebras
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