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 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 A p ( G ) with p ∈ ( 1 , ∞ ) . 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 ∈ ( 1 , ∞ ) : these are precisely the abelian groups.