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.