Connes Embeddings and von Neumann Regular Closures of Group Algebras

The analytic von Neumann regular closure $R(\mathrm{\Gamma })$ of a complex group algebra $\text{C}\mathrm{\Gamma }$ was introduced by Linnell and Schick. This ring is the smallest $\ast$-regular subring in the algebra of affiliated operators $U(\mathrm{\Gamma })$ containing $\text{C}\mathrm{\Gamma }$. We prove that all the algebraic von Neumann regular closures corresponding to sofic representations of an amenable group are isomorphic to $R(\mathrm{\Gamma })$. 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 $K\mathrm{\Gamma }$ over any field $K$ can be embedded to the rank completion of an ultramatricial algebra.