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Connes embeddings and von Neumann regular closures of amenable group algebras


Author: Gábor Elek
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
Published electronically: December 13, 2012
MathSciNet review: 3034457
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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
Email: elek@renyi.hu

DOI: https://doi.org/10.1090/S0002-9947-2012-05687-X
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
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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