Abstract view
Norm One Idempotent $cb$Multipliers with Applications to the Fourier Algebra in the $cb$Multiplier Norm


Published:20110520
Printed: Dec 2011
Brian E. Forrest,
Department of Pure Mathematics, University of Waterloo, Waterloo, ON, N2L 3G1
Volker Runde,
Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, T6G 2G1
Abstract
For a locally compact group $G$, let $A(G)$ be its Fourier algebra, let $M_{cb}A(G)$ denote the completely
bounded multipliers of $A(G)$, and let $A_{\mathit{Mcb}}(G)$ stand for the closure of $A(G)$ in $M_{cb}A(G)$. We
characterize the norm one idempotents in $M_{cb}A(G)$: the indicator function of a set $E \subset G$ is a norm
one idempotent in $M_{cb}A(G)$ if and only if $E$ is a coset of an open subgroup of $G$. As applications, we
describe the closed ideals of $A_{\mathit{Mcb}}(G)$ with an approximate identity bounded by $1$, and we characterize
those $G$ for which $A_{\mathit{Mcb}}(G)$ is $1$amenable in the sense of B. E. Johnson. (We can even slightly
relax the norm bounds.)
MSC Classifications: 
43A22, 20E05, 43A30, 46J10, 46J40, 46L07, 47L25 show english descriptions
Homomorphisms and multipliers of function spaces on groups, semigroups, etc. Free nonabelian groups Fourier and FourierStieltjes transforms on nonabelian groups and on semigroups, etc. 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. 20E05  Free nonabelian groups 43A30  Fourier and FourierStieltjes transforms on nonabelian groups and on semigroups, etc. 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]
