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Operator Amenability of the Fourier Algebra in the cb-Multiplier Norm

Published online by Cambridge University Press:  20 November 2018

Brian E. Forrest ,
Volker Runde  and
Nico Spronk
Brian E. Forrest
Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo, ON, N2L 3G1 email: beforres@math.uwaterloo.ca, nspronk@math.uwaterloo.ca
Volker Runde
Affiliation:
Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, T6G 2G1 email: vrunde@ualberta.ca
Nico Spronk
Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo, ON, N2L 3G1 email: beforres@math.uwaterloo.ca, nspronk@math.uwaterloo.ca
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Abstract

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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.

Keywords

43A2243A3046H2546J1046J4046L0747L25cb-multiplier normFourier algebraoperator amenabilityweak amenability
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 59 , Issue 5 , 01 October 2007 , pp. 966 - 980
Copyright
Copyright © Canadian Mathematical Society 2007

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