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