A note on ideal C^*-completions and amenability

For a discrete group G , we consider certain ideals I⊂ c₀(G) I ⊂ 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 C^*I(G) C I ∗ ( G ) in the sense of Brown and Guentner (Bull. London Math. Soc. 45:1181–1193, 2013) implies that G is amenable.