Operator Amenability of the Fourier Algebra in the cb-Multiplier Norm

Forrest, Brian E. ; Runde, Volker ; Spronk, Nico

Canadian journal of mathematics, Tome 59 (2007) no. 5, pp. 966-980

Résumé

Let $G$ be a locally compact group, and let ${{A}_{\text{cb}}}(G)$ denote the closure of $A(G)$ , the Fourier algebra of $G$ , in the space of completely bounded multipliers of $A(G)$ . If $G$ is a weakly amenable, discrete group such that ${{C}^{*}}(G)$ is residually finite-dimensional, we show that ${{A}_{\text{cb}}}(G)$ is operator amenable. In particular, ${{A}_{\text{cb}}}({{\mathbb{F}}_{2}})$ is operator amenable even though ${{\mathbb{F}}_{2}}$ , the free group in two generators, is not an amenable group. Moreover, we show that if $G$ is a discrete group such that ${{A}_{\text{cb}}}(G)$ is operator amenable, a closed ideal of $A(G)$ is weakly completely complemented in $A(G)$ if and only if it has an approximate identity bounded in the $\text{cb}$ -multiplier norm.

Détail

DOI : 10.4153/CJM-2007-041-9

Mots-clés : 43A22, 43A30, 46H25, 46J10, 46J40, 46L07, 47L25, cb-multiplier norm, Fourier algebra, operator amenability, weak amenability

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