Non-classification of free Araki-Woods factors and \(\tau \)-invariants (Q2286349)

Summary: 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 cocycle and outer conjugacy of actions of abelian groups on free product factors, are not classifiable by countable structures.