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# Operator Amenability of the Fourier Algebra in the $\cb$-Multiplier Norm

Published:2007-10-01
Printed: Oct 2007
• Brian E. Forrest
• Volker Runde
• Nico Spronk
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## Abstract

Let $G$ be a locally compact group, and let $A_{\cb}(G)$ 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 $\cstar(G)$ is residually finite-dimensional, we show that $A_{\cb}(G)$ is operator amenable. In particular, $A_{\cb}(\free_2)$ is operator amenable even though $\free_2$, the free group in two generators, is not an amenable group. Moreover, we show that if $G$ is a discrete group such that $A_{\cb}(G)$ 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 $\cb$-multiplier norm.
 Keywords: $\cb$-multiplier norm, Fourier algebra, operator amenability, weak amenability
 MSC Classifications: 43A22 - Homomorphisms and multipliers of function spaces on groups, semigroups, etc. 43A30 - Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc. 46H25 - Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX) 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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