The classification problem for von Neumann factors

We prove that it is not possible to classify separable von Neumann factors of types ${\text{II}}_1$, ${\text{II}}_{\mathrm{\infty }}$ or ${\text{III}}_{\lambda }$, $0\le \lambda \le 1$, up to isomorphism by a Borel measurable assignment of “countable structures” as invariants. In particular the isomorphism relation of type ${\text{II}}_1$ factors is not smooth. We also prove that the isomorphism relation for von Neumann ${\text{II}}_1$ factors is analytic, but is not Borel.