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Lamplighter groups and von Neumann`s continuous regular ring

Author: Gábor Elek
Journal: Proc. Amer. Math. Soc. 144 (2016), 2871-2883
MSC (2010): Primary 46L10
Published electronically: March 22, 2016
MathSciNet review: 3487221
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Abstract: Let $ \Gamma $ be a discrete group. Following Linnell and Schick one can define a continuous ring $ c(\Gamma )$ associated with $ \Gamma $. They proved that if the Atiyah Conjecture holds for a torsion-free group $ \Gamma $, then $ c(\Gamma )$ is a skew field. Also, if $ \Gamma $ has torsion and the Strong Atiyah Conjecture holds for $ \Gamma $, then $ c(\Gamma )$ is a matrix ring over a skew field. The simplest example when the Strong Atiyah Conjecture fails is the lamplighter group $ \Gamma =\mathbb{Z}_2\wr \mathbb{Z}$. It is known that $ \mathbb{C}(\mathbb{Z}_2\wr \mathbb{Z})$ does not even have a classical ring of quotients. Our main result is that if $ H$ is amenable, then $ c(\mathbb{Z}_2\wr H)$ is isomorphic to a continuous ring constructed by John von Neumann in the 1930s.

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Additional Information

Gábor Elek
Affiliation: Department of Mathematics and Statistics, Lancaster University, Lancaster, United Kingdom, LA1 4YF

Keywords: Continuous rings, von Neumann algebras, the algebra of affiliated operators, lamplighter group
Received by editor(s): April 26, 2014
Published electronically: March 22, 2016
Additional Notes: This research was partly sponsored by MTA Renyi “Lendulet” Groups and Graphs Research Group
Communicated by: Marius Junge
Article copyright: © Copyright 2016 American Mathematical Society

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