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Non-classification of Cartan subalgebras for a class of von Neumann algebras. (English) Zbl 1400.46049

Summary: We study the complexity of the classification problem for Cartan subalgebras in von Neumann algebras. We construct a large family of \(\mathrm{II}_{1}\) factors whose Cartan subalgebras up to unitary conjugacy are not classifiable by countable structures, providing the first such examples. Additionally, we construct examples of \(\mathrm{II}_{1}\) 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 \(\mathrm{II}_{1}\) factor up to unitary conjugacy are not classifiable by countable structures, and deduce that the same holds for any McDuff \(\mathrm{II}_{1}\) factor with at least one Cartan subalgebra.

MSC:

46L36 Classification of factors
03E15 Descriptive set theory
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