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

Abstract

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