Majorations Uniformes de Normes D'Inverses Dans Les Algèbres de Beurling

The Beurling algebras l 1 ( D ,ω)( D =N,Z) that are semi‐simple, with compact Gelfand transform, are considered. The paper gives a necessary and sufficient condition (on ω) such that l 1 ( D ,ω) possesses a uniform quantitative version of Wiener's theorem in the sense that there exists a function ϕ:]0,+∞[→]0,+∞ such that, for every invertible element x in the unit ball of l 1 ( D ,ω), we have ‖ x −1 ‖⩽ϕ( r ( x −1 )) r ( x −1 ) is the spectral radius of x −1 .