A note on ideal C\(^*\)-completions and amenability

For a discrete group G, we consider certain ideals \(\mathcal {I}\subset c_0(G)\) $\mathcal{I}\subset c_0(G)$ 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 \(\textrm{C}^*_{\mathcal {I}}(G)\) ${\text{C}}_{\mathcal{I}}^{\ast }(G)$ in the sense of Brown and Guentner (Bull. London Math. Soc. 45:1181–1193, 2013) implies that G is amenable.