Cohen–Host type idempotent theorems for representations on Banach spaces and applications to Figà-Talamanca–Herz algebras

Let G be a locally compact group, and let R(G) denote the ring of subsets of G generated by the left cosets of open subsets of G. The Cohen–Host idempotent theorem asserts that a set lies in R(G) if and only if its indicator function is a coefficient function of a unitary representation of G on some Hilbert space. We prove related results for representations of G on certain Banach spaces. We apply our Cohen–Host type theorems to the study of the Figà-Talamanca–Herz algebras Ap(G) with p∈(1,∞). For arbitrary G, we characterize those closed ideals of Ap(G) that have an approximate identity bounded by 1 in terms of their hulls. Furthermore, we characterize those G such that Ap(G) is 1-amenable for some—and, equivalently, for all—p∈(1,∞): these are precisely the abelian groups.