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

Forrest, Brian E. ; Runde, Volker

Canadian mathematical bulletin, Tome 54 (2011) no. 4, pp. 654-662

Résumé

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.)

Détail

DOI : 10.4153/CMB-2011-098-0

Mots-clés : 43A22, 20E05, 43A30, 46J10, 46J40, 46L07, 47L25, amenability, bounded approximate identity, cb-multiplier norm, Fourier algebra, norm one idempotent