A note on ideal C $$^*$$ -completions and amenability

Abstract

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\(^*\)-algebra of G and the C\(^*\)-completion \(\textrm{C}^*_{\mathcal {I}}(G)\) in the sense of Brown and Guentner (Bull. London Math. Soc. 45:1181–1193, 2013) implies that G is amenable.