On group rings of virtually abelian groups

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