Idempotents in quotients and restrictions of Banach algebras of functions

Pedersen, Thomas Vils

Annales de l'Institut Fourier, Tome 46 (1996) no. 4, pp. 1095-1124.

Soit ${𝒜}_{\beta }$ l’algèbre de Beurling à poids $(1+|n|{)}^{\beta }$ sur le cercle unité $𝕋$ et, pour un ensemble fermé $E\subseteq 𝕋$, soit $J_{{𝒜}_{\beta }}\left(E\right)=\{f\in {𝒜}_{\beta }:f=0\text{au}\text{voisinage}\text{de}E\}$. 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.

Let ${𝒜}_{\beta }$ be the Beurling algebra with weight $(1+|n|{)}^{\beta }$ on the unit circle $𝕋$ and, for a closed set $E\subseteq 𝕋$, let $J_{{𝒜}_{\beta }}\left(E\right)=\{f\in {𝒜}_{\beta }:f=0\text{on}\text{a}\text{neighbourhood}\text{of}E\}$. 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.

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