Connes Embeddings and von Neumann Regular Closures of Group Algebras

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.