Separable II₁ factors with property Gamma and full II₁ factors cannot be classified up to isomorphism by Borel measurable assignments of countable structures.
Vaes, An inner amenable group whose von Neumann algebra does not ha ve property Gamma, Acta Mathematica 208 (2012), 389–394
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A note on the classification of Gamma factors
Separable II₁ factors with property Gamma and full II₁ factors cannot be classified up to isomorphism by Borel measurable assignments of countable structures.