Lamplighter groups and von Neumann‘s continuous regular ring

Abstract

Let Γ \Gamma be a discrete group. Following Linnell and Schick one can define a continuous ring c ( Γ ) c(\Gamma ) associated with Γ \Gamma . They proved that if the Atiyah Conjecture holds for a torsion-free group Γ \Gamma , then c ( Γ ) c(\Gamma ) is a skew field. Also, if Γ \Gamma has torsion and the Strong Atiyah Conjecture holds for Γ \Gamma , then c ( Γ ) c(\Gamma ) is a matrix ring over a skew field. The simplest example when the Strong Atiyah Conjecture fails is the lamplighter group Γ = Z 2 ≀ Z \Gamma =\mathbb {Z}_2\wr \mathbb {Z} . It is known that C ( Z 2 ≀ Z ) \mathbb {C}(\mathbb {Z}_2\wr \mathbb {Z}) does not even have a classical ring of quotients. Our main result is that if H H is amenable, then c ( Z 2 ≀ H ) c(\mathbb {Z}_2\wr H) is isomorphic to a continuous ring constructed by John von Neumann in the 1930s.