Connes Embeddings and von Neumann Regular Closures of Group Algebras

Details Connes Embeddings and von Neumann Regular Closures of Group Algebras Connes Embeddings and von Neumann Regular Closures of Group Algebras

2010-06-28

arxiv.org/abs/1006.5378v1

Mathematics

Operator Algebras

Scientific paper

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.

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