Submission date: 24 March 2024

Abstract:

We prove that if G and H are α-back-and-forth equivalent groups (in the sense of computable structure theory) for some ordinal α ≥ ω, then their group von Neumann algebras L(G) and L(H) are also α-back-and-forth equivalent. In particular, if G and H are ω-back-and-forth-equivalent groups, then L(G) and L(H) are elementarily equivalent; this is known to fail under the weaker hypothesis that G and H are merely elementarily equivalent. We extend this result to crossed product von Neumann algebras associated to Bernoulli actions of back-and-forth equivalent groups.

Mathematics Subject Classification:

Keywords and phrases:

Full text arXiv 2403.16181: pdf, ps.