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Operator Amenability of the Fourier Algebra in the $\cb$-Multiplier Norm |
Canad. J. Math. 59(2007), 966-980
Published:2007-10-01
Printed: Oct 2007
Printed: Oct 2007
- Brian E. Forrest
- Volker Runde
- Nico Spronk
Abstract
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 amenability |
| MSC Classifications: |
43A22 - Homomorphisms and multipliers of function spaces on groups, semigroups, etc. 43A30 - Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc. 46H25 - Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX) 46J10 - Banach algebras of continuous functions, function algebras [See also 46E25] 46J40 - Structure, classification of commutative topological algebras 46L07 - Operator spaces and completely bounded maps [See also 47L25] 47L25 - Operator spaces (= matricially normed spaces) [See also 46L07] |