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

We study the complexity of the classification problem for Cartan subalgebras in von Neumann algebras. We construct a large family of II1 factors whose Cartan subalgebras up to unitary conjugacy are not classifiable by countable structures, providing the first such examples. Additionally, we construct examples of II1 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 II1 factor up to unitary conjugacy are not classifiable by countable structures, and deduce that the same holds for any McDuff II1 factor with at least one Cartan subalgebra.