Idempotents in quotients and restrictions of Banach algebras of functions

Pedersen, Thomas Vils

Annales de l'Institut Fourier, Tome 46 (1996), p. 1095-1124 / Harvested from Numdam

Résumé

Soit ${\cal A}_\beta $ l’algèbre de Beurling à poids $(1+\vert n\vert )^\beta $ sur le cercle unité ${\Bbb T}$ et, pour un ensemble fermé $E\subseteq {\Bbb T}$, soit $J_{{\cal A}_\beta }(E)=\lbrace f\in {\cal A}_\beta :f=0\,\text{au} \text{voisinage} \text{de}\,E\rbrace $. Nous montrons que, pour $\beta >{1\over 2}$, il existe un ensemble fermé $E\subseteq {\Bbb T}$ de mesure nulle tel que l’algèbre quotient ${\cal A}_\beta /\overline{J_{{\cal A}_\beta }(E)}$ n’est pas engendrée par ses idempotents, contrastant par là avec un résultat de Zouakia. De plus, pour les algèbres de Lipschitz $\lambda _\gamma $ et l’algèbre ${\cal A}{\cal C}$ des fonctions absolument continues sur ${\Bbb T}$, nous caractérisons les ensembles fermés $E\subseteq {\Bbb T}$ tels que les algèbres restrictions $\lambda _\gamma (E)$ et ${\cal A}{\cal C} (E)$ soient engendrées par leurs idempotents.

Abstract

Let ${\cal A}_\beta $ be the Beurling algebra with weight $(1+\vert n\vert )^\beta $ on the unit circle ${\Bbb T}$ and, for a closed set $E\subseteq {\Bbb T}$, let $J_{{\cal A}_\beta }(E)=\lbrace f\in {\cal A}_\beta :f=0\,\text{on} \text{a} \text{neighbourhood} \text{of}\, E\rbrace $. We prove that, for $\beta >{1\over 2}$, there exists a closed set $E\subseteq {\Bbb T}$ of measure zero such that the quotient algebra ${\cal A}_\beta /\overline{J_{{\cal A}_\beta }(E)}$ is not generated by its idempotents, thus contrasting a result of Zouakia. Furthermore, for the Lipschitz algebras $\lambda _\gamma $ and the algebra ${\cal A}{\cal C}$ of absolutely continuous functions on ${\Bbb T}$, we characterize the closed sets $E\subseteq {\Bbb T}$ for which the restriction algebras $\lambda _\gamma (E)$ and ${\cal A}{\cal C} (E)$ are generated by their idempotents.

Détail

Publié le : 1996-01-01

MR 98b:46070 | Zbl 0853.46047

DOI : https://doi.org/10.5802/aif.1542

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