Operator amenability of the Fourier algebra in the cb-multiplier norm

Details Operator amenability of the Fourier algebra in the cb-multiplier norm Operator amenability of the Fourier algebra in the cb-multiplier norm

2005-01-06

arxiv.org/abs/math/0501092v3

Canadian J. Math. 59 (2007), 966-980

Mathematics

Functional Analysis

LaTeX2e; 18 pages; cleaned up a bit

Scientific paper

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, $A_\cb(F_2)$ is operator amenable even though $F_2$, the free group in two generators, is not an amenable group. Moreover, we show that, if $G$ is a discrete group such that $A_\cb(G)$ 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.

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