CMS/SMC
Canadian Mathematical Society
www.cms.math.ca
Canadian Mathematical Society
  location:  PublicationsjournalsCMB
Publications        
Abstract view

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

  Published:2011-05-20
 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
Format:   LaTeX   MathJax   PDF  

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.)
Keywords: amenability, bounded approximate identity, $cb$-multiplier norm, Fourier algebra, norm one idempotent amenability, bounded approximate identity, $cb$-multiplier norm, Fourier algebra, norm one idempotent
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 Fourier-Stieltjes 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 Fourier-Stieltjes 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]
 

© Canadian Mathematical Society, 2014 : https://cms.math.ca/