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

TL;DR

This paper demonstrates that free Araki-Woods factors and related invariants like $ au$-topologies are not classifiable by countable structures, highlighting complexity in their classification.

Contribution

It introduces the standard Borel space for free Araki-Woods factors and proves their isomorphism relation is not classifiable by countable structures.

Findings

Isomorphism relation of free Araki-Woods factors is not classifiable by countable structures.

Equality of $ au$-topologies is not classifiable by countable structures.

Cocycle and outer conjugacy of actions on free product factors are not classifiable by countable structures.

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.