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Operator Amenability of the Fourier Algebra in the cb-Multiplier Norm
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- Journal:
- Canadian Journal of Mathematics / Volume 59 / Issue 5 / 01 October 2007
- Published online by Cambridge University Press:
- 20 November 2018, pp. 966-980
- Print publication:
- 01 October 2007
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- Article
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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.
The Operator Amenability of Uniform Algebras
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- Journal:
- Canadian Mathematical Bulletin / Volume 46 / Issue 4 / 01 December 2003
- Published online by Cambridge University Press:
- 20 November 2018, pp. 632-634
- Print publication:
- 01 December 2003
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- Article
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We prove a quantized version of a theorem by M. V. Sheĭnberg: A uniform algebra equipped with its canonical, i.e., minimal, operator space structure is operator amenable if and only if it is a commutative
${{C}^{*}}$ -algebra.
