On group rings of virtually abelian groups

Abstract

Let $Γ$ be a finitely generated torsion-free group. We show that the statement of $Γ$ being virtually abelian is equivalent to the statement that the $*$-regular closure of the group ring $\mathbb{C}[Γ]$ in the algebra of (unbounded) operators affiliated to the group von Neumann algebra is a central division algebra. More generally, for any field $k$, it is shown that $k[Γ]$ embeds into a central division algebra in case $Γ$ is virtually abelian. We take advantage of this result in order to develop a criterion for existence of units in the group ring $k[Γ]$. We develop this criterion in the particular case of $Γ$ being the Promislow's group.