Idempotents in quotients and restrictions of Banach algebras of functions
Annales de l'Institut Fourier, Volume 46 (1996) no. 4, pp. 1095-1124.

Let ${𝒜}_{\beta }$ be the Beurling algebra with weight $\left(1+|n|{\right)}^{\beta }$ on the unit circle $𝕋$ and, for a closed set $E\subseteq 𝕋$, let ${J}_{{𝒜}_{\beta }}\left(E\right)=\left\{f\in {𝒜}_{\beta }:f=0\phantom{\rule{0.166667em}{0ex}}\text{on}\phantom{\rule{4pt}{0ex}}\text{a}\phantom{\rule{4pt}{0ex}}\text{neighbourhood}\phantom{\rule{4pt}{0ex}}\text{of}\phantom{\rule{0.166667em}{0ex}}E\right\}$. We prove that, for $\beta >\frac{1}{2}$, there exists a closed set $E\subseteq 𝕋$ of measure zero such that the quotient algebra ${𝒜}_{\beta }/\overline{{J}_{{𝒜}_{\beta }}\left(E\right)}$ is not generated by its idempotents, thus contrasting a result of Zouakia. Furthermore, for the Lipschitz algebras ${\lambda }_{\gamma }$ and the algebra $𝒜𝒞$ of absolutely continuous functions on $𝕋$, we characterize the closed sets $E\subseteq 𝕋$ for which the restriction algebras ${\lambda }_{\gamma }\left(E\right)$ and $𝒜𝒞\left(E\right)$ are generated by their idempotents.

Soit ${𝒜}_{\beta }$ l’algèbre de Beurling à poids $\left(1+|n|{\right)}^{\beta }$ sur le cercle unité $𝕋$ et, pour un ensemble fermé $E\subseteq 𝕋$, soit ${J}_{{𝒜}_{\beta }}\left(E\right)=\left\{f\in {𝒜}_{\beta }:f=0\phantom{\rule{0.166667em}{0ex}}\text{au}\phantom{\rule{4pt}{0ex}}\text{voisinage}\phantom{\rule{4pt}{0ex}}\text{de}\phantom{\rule{0.166667em}{0ex}}E\right\}$. Nous montrons que, pour $\beta >\frac{1}{2}$, il existe un ensemble fermé $E\subseteq 𝕋$ de mesure nulle tel que l’algèbre quotient ${𝒜}_{\beta }/\overline{{J}_{{𝒜}_{\beta }}\left(E\right)}$ 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 $𝒜𝒞$ des fonctions absolument continues sur $𝕋$, nous caractérisons les ensembles fermés $E\subseteq 𝕋$ tels que les algèbres restrictions ${\lambda }_{\gamma }\left(E\right)$ et $𝒜𝒞\left(E\right)$ soient engendrées par leurs idempotents.

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author = {Pedersen, Thomas Vils},
title = {Idempotents in quotients and restrictions of {Banach} algebras of functions},
journal = {Annales de l'Institut Fourier},
pages = {1095--1124},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {46},
number = {4},
year = {1996},
doi = {10.5802/aif.1542},
mrnumber = {98b:46070},
zbl = {0853.46047},
language = {en},
url = {http://www.numdam.org/articles/10.5802/aif.1542/}
}
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Pedersen, Thomas Vils. Idempotents in quotients and restrictions of Banach algebras of functions. Annales de l'Institut Fourier, Volume 46 (1996) no. 4, pp. 1095-1124. doi : 10.5802/aif.1542. http://www.numdam.org/articles/10.5802/aif.1542/

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