Connes Embeddings and von Neumann Regular Closures of Group Algebras

Abstract

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