Back-and-forth equivalent group von Neumann algebras

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 $α\geq ω$, 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.

Theorems & Definitions (63)

• Theorem 1.2
• Theorem 1.2
• Theorem 1.2
• Definition 2.1
• Definition 2.3
• Remark 2.4
• Remark 2.5
• Theorem 2.6: Scott Scott
• Definition 2.7
• Definition 2.8