Turbulence and Araki-Woods factors

TL;DR

This paper demonstrates that Araki-Woods factors and certain odometer actions are not classifiable by countable structures, using Baire category techniques to strengthen known results in operator algebra classification.

Contribution

It introduces new Baire category methods to prove non-classifiability of Araki-Woods factors and provides a new proof that ITPFI factors' isomorphism problem is not smooth.

Findings

Araki-Woods factors are not classifiable by countable structures.

The isomorphism problem for ITPFI factors is not smooth.

Odometer actions of Z are not classifiable up to orbit equivalence by countable structures.

Abstract

Using Baire category techniques we prove that Araki-Woods factors are not classifiable by countable structures. As a result, we obtain a far reaching strengthening as well as a new proof of the well-known theorem of Woods that the isomorphism problem for ITPFI factors is not smooth. We derive as a consequence that the odometer actions of Z that preserve the measure class of a finite non-atomic product measure are not classifiable up to orbit equivalence by countable structures.