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Operator Amenability of the Fourier Algebra in the cb-Multiplier Norm
Published online by Cambridge University Press: 20 November 2018
Abstract
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.
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- Copyright © Canadian Mathematical Society 2007
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