Non-classification of Cartan subalgebras for a class of von Neumann algebras

TL;DR

This paper demonstrates the complexity of classifying Cartan subalgebras in certain von Neumann algebras, showing they are not classifiable by countable structures, including hyperfinite and McDuff II$_1$ factors.

Contribution

It constructs the first examples of II$_1$ factors with non-classifiable Cartan subalgebras under conjugacy and automorphism, advancing understanding of their classification complexity.

Findings

Cartan subalgebras in constructed II$_1$ factors are not classifiable by countable structures.

Hyperfinite II$_1$ factor's Cartan subalgebras are not classifiable.

McDuff II$_1$ factors with at least one Cartan subalgebra also have non-classifiable Cartan subalgebras.

Abstract

We study the complexity of the classification problem for Cartan subalgebras in von Neumann algebras. We construct a large family of II$_1$ factors whose Cartan subalgebras up to unitary conjugacy are not classifiable by countable structures, providing the first such examples. Additionally, we construct examples of II$_1$ factors whose Cartan subalgebras up to conjugacy by an automorphism are not classifiable by countable structures. Finally, we show directly that the Cartan subalgebras of the hyperfinite II$_1$ factor up to unitary conjugacy are not classifiable by countable structures, and deduce that the same holds for any McDuff II$_1$ factor with at least one Cartan subalgebra.