The classification problem for von Neumann factors

We prove that it is not possible to classify separable von Neumann factors of types II1, II∞ or IIIλ, 0⩽λ⩽1, up to isomorphism by a Borel measurable assignment of “countable structures” as invariants. In particular the isomorphism relation of type II1 factors is not smooth. We also prove that the isomorphism relation for von Neumann II1 factors is analytic, but is not Borel.