Idempotents in quotients and restrictions of Banach algebras of functions
Annales de l'Institut Fourier, Volume 46 (1996) no. 4, p. 1095-1124
Let 𝒜 β be the Beurling algebra with weight (1+|n|) β on the unit circle 𝕋 and, for a closed set E𝕋, let J 𝒜 β (E)={f𝒜 β :f=0onaneighbourhoodofE}. We prove that, for β>1 2, there exists a closed set E𝕋 of measure zero such that the quotient algebra 𝒜 β /J 𝒜 β (E) ¯ is not generated by its idempotents, thus contrasting a result of Zouakia. Furthermore, for the Lipschitz algebras λ γ and the algebra 𝒜𝒞 of absolutely continuous functions on 𝕋, we characterize the closed sets E𝕋 for which the restriction algebras λ γ (E) and 𝒜𝒞(E) are generated by their idempotents.
Soit 𝒜 β l’algèbre de Beurling à poids (1+|n|) β sur le cercle unité 𝕋 et, pour un ensemble fermé E𝕋, soit J 𝒜 β (E)={f𝒜 β :f=0auvoisinagedeE}. Nous montrons que, pour β>1 2, il existe un ensemble fermé E𝕋 de mesure nulle tel que l’algèbre quotient 𝒜 β /J 𝒜 β (E) ¯ n’est pas engendrée par ses idempotents, contrastant par là avec un résultat de Zouakia. De plus, pour les algèbres de Lipschitz λ γ et l’algèbre 𝒜𝒞 des fonctions absolument continues sur 𝕋, nous caractérisons les ensembles fermés E𝕋 tels que les algèbres restrictions λ γ (E) et 𝒜𝒞(E) soient engendrées par leurs idempotents.
@article{AIF_1996__46_4_1095_0,
     author = {Pedersen, Thomas Vils},
     title = {Idempotents in quotients and restrictions of Banach algebras of functions},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {46},
     number = {4},
     year = {1996},
     pages = {1095-1124},
     doi = {10.5802/aif.1542},
     zbl = {0853.46047},
     mrnumber = {98b:46070},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1996__46_4_1095_0}
}
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/item/AIF_1996__46_4_1095_0/

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