Lamplighter groups and von neumann‘s continuous regular ring

Elek, Gábor (2016) Lamplighter groups and von neumann‘s continuous regular ring. PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 144 (7). 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 1930s. © 2016 American Mathematical Society.

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