A note on ideal C${}^{\ast }$-completions and amenability

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.