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

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 \[ \|x^{-1}\|\le \phi(r(x^{-1}))\quad r(x^{-1})\hbox{ is the spectral radius of }x^{-1}. \]