Lamplighter groups and von Neumann's continuous regular rings

Elek, Gabor (2016) Lamplighter groups and von Neumann's continuous regular rings. Proceedings of the American Mathematical Society, 144. pp. 2871-2883. ISSN 0002-9939

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Abstract

Let Γ be a discrete group. Following Linnell and Schick one can define a continuous ring c(Γ) associated with Γ. They proved that if the Atiyah Conjecture holds for a torsion-free group Γ, then c(Γ) is a skew field. Also, if Γ has torsion and the Strong Atiyah Conjecture holds for Γ, then c(Γ) is a matrix ring over a skew field. The simplest example when the Strong Atiyah Conjecture fails is the lamplighter group Γ = Z2 ≀ Z. It is known that C(Z2 ≀ Z) does not even have a classical ring of quotients. Our main result is that if H is amenable, then c(Z2 ≀H) is isomorphic to a continuous ring constructed by John von Neumann in the 1930′s.

Item Type:
Journal Article
Journal or Publication Title:
Proceedings of the American Mathematical Society
Additional Information:
First published in Proceedings of the American Mathematical Society in 144 (2016), published by the American Mathematical Society
Uncontrolled Keywords:
/dk/atira/pure/subjectarea/asjc/2600
Subjects:
ID Code:
77404
Deposited By:
Deposited On:
20 Jan 2016 16:22
Refereed?:
Yes
Published?:
Published
Last Modified:
03 Jul 2020 02:04