Non-classification of free Araki-Woods factors and $\tau$-invariants

Abstract

We define the standard Borel space of free Araki-Woods factors and prove that their isomorphism relation is not classifiable by countable structures. We also prove that equality of $\tau$-topologies, arising as invariants of type III factors, as well as coycle and outer conjugacy of actions of abelian groups on free product factors are not classifiable by countable structures.

arXiv commentv3: minor changes, final version, to appear in Groups, Geometry, and Dynamics