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    Rights statement: First published in Proceedings of the American Mathematical Society in 144 (2016), published by the American Mathematical Society

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Lamplighter groups and von Neumann's continuous regular rings

Research output: Contribution to journalJournal article

Published
<mark>Journal publication date</mark>2016
<mark>Journal</mark>Proceedings of the American Mathematical Society
Volume144
Number of pages13
Pages (from-to)2871-2883
Publication statusPublished
Early online date22/03/16
Original languageEnglish

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.

Bibliographic note

First published in Proceedings of the American Mathematical Society in 144 (2016), published by the American Mathematical Society