The classification problem for von Neumann factors

2009/03/25 by Román Sasyk, Sasyk, Roman, Asger Törnquist +1
Mathematics · #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Logic (math.LO) #Operator Algebras (math.OA) #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.0903.4484

openalex publication_date 2009/03/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that it is not possible to classify separable von Neumann factors of types \II₁, \II_∞ or \III_λ, 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.

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