Connes Embeddings and von Neumann Regular Closures of Group Algebras

TL;DR

This paper investigates the structure of von Neumann regular closures of group algebras, showing their isomorphism for sofic representations of amenable groups and connecting to Lück's Approximation Theorem.

Contribution

It demonstrates that all algebraic von Neumann regular closures from sofic representations of an amenable group are isomorphic to the analytic closure, generalizing existing theorems.

Findings

All algebraic von Neumann regular closures are isomorphic to the analytic closure R(Γ).

Amenable group algebras embed into the rank completion of ultramatricial algebras.

The result extends Lück's Approximation Theorem to a broader context.

Abstract

The analytic von Neumann regular closure $R(\Gamma)$ of a complex group algebra $\C\Gamma$ was introduced by Linnell and Schick. This ring is the smallest $*$-regular subring in the algebra of affiliated operators $U(\Gamma)$ containing $\C\Gamma$. We prove that all the algebraic von Neumann regular closures corresponding to sofic representations of an amenable group are isomorphic to $R(\Gamma)$. This result can be viewed as a structural generalization of L\"uck's Approximation Theorem. \noindent The main tool of the proof which might be of independent interest is that an amenable group algebra $K\Gamma$ over any field $K$ can be embedded to the rank completion of an ultramatricial algebra.