On group rings of virtually abelian groups

Abstract

Let $\Gamma$ be a finitely generated torsion-free group. We show that the statement of $\Gamma$ being virtually abelian is equivalent to the statement that the $\ast$-regular closure of the group ring $\mathbb{C}[\Gamma ]$ 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 $k[\Gamma ]$ embeds into a central division algebra in case $\Gamma$ is virtually abelian. We take advantage of this result in order to develop a criterion for existence of units in the group ring $k[\Gamma ]$. We develop this criterion in the particular case of $\Gamma$ being the Promislow's group.

24 pages