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

Abstract

Let $G$ be a locally compact group, and let $\mathcal{R}(G)$ 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 $\mathcal{R}(G)$ 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 exchanged