TY - JOUR

T1 - Lamplighter groups and von Neumann's continuous regular rings

AU - Elek, Gabor

N1 - First published in Proceedings of the American Mathematical Society in 144 (2016), published by the American Mathematical Society

PY - 2016

Y1 - 2016

N2 - 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.

AB - 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.

KW - continuous rings

KW - discrete groups

U2 - 10.1090/proc/13066

DO - 10.1090/proc/13066

M3 - Journal article

VL - 144

SP - 2871

EP - 2883

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

SN - 0002-9939

ER -