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MAJORATIONS UNIFORMES DE NORMES D'INVERSES DANS LES ALGÈBRES DE BEURLING

Published online by Cambridge University Press:  24 March 2003

O. EL-FALLAH
Affiliation:
Departement de Mathématiques et Informatique, Faculté des Sciences, Université Mohammed V, Avenue Ibn Battouta, BP 1014, Rabat, [email protected]
A. EZZAARAOUI
Affiliation:
Departement de Mathématiques et Informatique, Faculté des Sciences, Université Mohammed V, Avenue Ibn Battouta, BP 1014, Rabat, [email protected]
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Abstract

The Beurling algebras $l^1({\cal D},\omega)\;({\cal D}={\bb N},{\bb Z})$ that are semi-simple, with compact Gelfand transform, are considered. The paper gives a necessary and sufficient condition (on $\omega$ ) such that $l^1({\cal D},\omega)$ possesses a uniform quantitative version of Wiener's theorem in the sense that there exists a function $\phi:]0,+\infty[\longrightarrow ]0,+\infty[$ such that, for every invertible element $x$ in the unit ball of $l^1({\cal D},\omega)$ , we have \[ \|x^{-1}\|\le \phi(r(x^{-1}))\quad r(x^{-1})\hbox{ is the spectral radius of }x^{-1}. \]

Type
Research Article
Copyright
The London Mathematical Society, 2002

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