Connes embeddings and von Neumann regular closures of amenable group algebras
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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.
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Additional Information
- Gábor Elek
- Affiliation: The Alfred Renyi Institute of Mathematics, Hungarian Academy of Sciences, Budapest, Hungary
- MR Author ID: 360750
- Email: elek@renyi.hu
- Received by editor(s): July 6, 2010
- Received by editor(s) in revised form: March 15, 2011, May 3, 2011, and August 21, 2011
- Published electronically: December 13, 2012
- Additional Notes: This research was sponsored by OTKA Grant No. 69062
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 365 (2013), 3019-3039
- MSC (2010): Primary 16S34, 22D25
- DOI: https://doi.org/10.1090/S0002-9947-2012-05687-X
- MathSciNet review: 3034457