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# Norm One Idempotent $cb$-Multipliers with Applications to the Fourier Algebra in the $cb$-Multiplier Norm

Published:2011-05-20
Printed: Dec 2011
• Brian E. Forrest,
Department of Pure Mathematics, University of Waterloo, Waterloo, ON, N2L 3G1
• Volker Runde,
Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, T6G 2G1
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## 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 $A_{\mathit{Mcb}}(G)$ 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 $A_{\mathit{Mcb}}(G)$ with an approximate identity bounded by $1$, and we characterize those $G$ for which $A_{\mathit{Mcb}}(G)$ is $1$-amenable in the sense of B. E. Johnson. (We can even slightly relax the norm bounds.)
 Keywords: amenability, bounded approximate identity, $cb$-multiplier norm, Fourier algebra, norm one idempotent
 MSC Classifications: 43A22 - Homomorphisms and multipliers of function spaces on groups, semigroups, etc. 20E05 - Free nonabelian groups 43A30 - Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc. 46J10 - Banach algebras of continuous functions, function algebras [See also 46E25] 46J40 - Structure, classification of commutative topological algebras 46L07 - Operator spaces and completely bounded maps [See also 47L25] 47L25 - Operator spaces (= matricially normed spaces) [See also 46L07]

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