Connes embeddings and von Neumann regular closures of amenable group algebras

Abstract:

The analytic von Neumann regular closure $R(\Gamma )$ of a complex group algebra $\mathbb {C}\Gamma$ was introduced by Linnell and Schick. This ring is the smallest $*$-regular subring in the algebra of affiliated operators $U(\Gamma )$ containing $\mathbb {C}\Gamma$. 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 Lück’s approximation theorem.

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.