2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/123304Let $G$ be a locally compact group, and let ${\cal R}(G)$ 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 ${\cal R}(G)$ 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 $A_p(G)$ with $p \in (1,\infty)$. For arbitrary $G$, we characterize those closed ideals of $A_p(G)$ that have an approximate identity bounded by 1 in terms of their hulls. Furthermore, we characterize those $G$ such that $A_p(G)$ is 1-amenable for some -- and, equivalently, for all -- $p \in (1,\infty)$: these are precisely the abelian groups.20 pages; LaTeX2e; one reference exchangedFunctional AnalysisPrimary 22D12; Secondary 22D05, 22D10, 43A07, 43A15, 43A30, 43A65, 46B08, 46B20, 46H20, 46H25, 46J10, 46J20, 46J40text