  For a discrete group $G$, we consider certain ideals $\mathcal{I}\subset
c_0(G)$ of sequences with prescribed rate of convergence to zero. We show that
the equality between the full group C$^\ast$-algebra of $G$ and the
C$^\ast$-completion $\mathrm{C}_{\mathcal{I}}^\ast(G)$ in the sense of Brown
and Guentner implies that $G$ is amenable.
