Norm one idempotent cb-multipliers with applications to the Fourier algebra in the cb-multiplier norm

TL;DR

This paper characterizes norm one idempotent multipliers in the Fourier algebra's cb-multiplier space for locally compact groups, revealing their structure and applications to ideals and amenability.

Contribution

It provides a complete characterization of norm one idempotents in cb-multiplier spaces and explores their implications for ideals and amenability in Fourier algebras.

Findings

Indicator functions of cosets of open subgroups are the only norm one idempotents.

Describes closed ideals with bounded approximate identities in A_{Mcb}(G).

Characterizes groups for which A_{Mcb}(G) is 1-amenable.

Abstract

For a locally compact group $G$, let $A(G)$ be its Fourier algebra, let $M_{cb}A(G)$ denote the completely bounded multipliers of $A(G)$, and let $A_{Mcb}(G)$ stand for the closure of $A(G)$ in $M_{cb}A(G)$. We characterize the norm one idempotents in $M_{cb}A(G)$: the indicator function of a set $E \subset G$ is a norm one idempotent in $M_{cb}A(G)$ if and only if $E$ is a coset of an open subgroup of $G$. As applications, we describe the closed ideals of $A_{Mcb}(G)$ with an approximate identity bounded by 1, and we characterize those $G$ for which $A_{Mcb}(G)$ is 1-amenable in the sense of B. E. Johnson. (We can even slightly relax the norm bounds.)