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1. CJM 2007 (vol 59 pp. 966)

Forrest, Brian E.; Runde, Volker; Spronk, Nico
 Operator Amenability of the Fourier Algebra in the \$\cb\$-Multiplier Norm 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 amenabilityCategories:43A22, 43A30, 46H25, 46J10, 46J40, 46L07, 47L25