Abstract view
Operator Amenability of the Fourier Algebra in the $\cb$Multiplier Norm


Published:20071001
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 finitedimensional, 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.
MSC Classifications: 
43A22, 43A30, 46H25, 46J10, 46J40, 46L07, 47L25 show english descriptions
Homomorphisms and multipliers of function spaces on groups, semigroups, etc. Fourier and FourierStieltjes transforms on nonabelian groups and on semigroups, etc. Normed modules and Banach modules, topological modules (if not placed in 13XX or 16XX) Banach algebras of continuous functions, function algebras [See also 46E25] Structure, classification of commutative topological algebras Operator spaces and completely bounded maps [See also 47L25] Operator spaces (= matricially normed spaces) [See also 46L07]
43A22  Homomorphisms and multipliers of function spaces on groups, semigroups, etc. 43A30  Fourier and FourierStieltjes transforms on nonabelian groups and on semigroups, etc. 46H25  Normed modules and Banach modules, topological modules (if not placed in 13XX or 16XX) 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]
