Connes embeddings and von Neumann regular closures of amenable group algebras (Q2838114)

The analytic von Neumann regular closure \(R(\Gamma)\) of a complex group algebra \(\mathbb{C}\Gamma\) is the smallest \(\star\)-regular subring containing \(\mathbb{C}\Gamma\) in the algebra of affiliated operators \(U(\Gamma)\). The author proves that all the algebraic von Neumann regular closures corresponding to sofic representations of an amenable group are isomorphic to \(R(\Gamma)\).NEWLINENEWLINE This result is a generalization of \textit{W. Lück}'s approximation theorem [Geom. Funct. Anal. 4, No. 4, 455--481 (1994; Zbl 0853.57021)].NEWLINENEWLINE The main tool used in the proof is that an amenable group algebra \(K\Gamma\) over any field \(K\) can be embedded to the rank completion of an ultramatricial algebra.