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

Published online by Cambridge University Press:  20 November 2018

Brian E. Forrest  and
Volker Runde
Brian E. Forrest
Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo, ON, N2L 3G1e-mail: beforres@math.uwaterloo.ca
Volker Runde
Affiliation:
Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, T6G 2G1e-mail: vrunde@ualberta.ca
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Abstract

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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.)

Keywords

43A2220E0543A3046J1046J4046L0747L25amenabilitybounded approximate identitycb-multiplier normFourier algebranorm one idempotent
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 54 , Issue 4 , 01 December 2011 , pp. 654 - 662
Copyright
Copyright © Canadian Mathematical Society 2011

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