A torsion-free algebraically $\mathrm{C}^*$-unique group

Abstract

Let $p$ and $q$ be multiplicatively independent integers. We show that the complex group ring of $\mathbb{Z}\left[1∕pq\right]⋊{\mathbb{Z}}^2$ admits a unique $C^{\ast }$-norm. The proof uses a characterization, due to Furstenberg, of closed $\times p$- and $\times q$-invariant subsets of $\mathbb{𝕋}$.