Let be the Beurling algebra with weight on the unit circle and, for a closed set , let . We prove that, for , there exists a closed set of measure zero such that the quotient algebra is not generated by its idempotents, thus contrasting a result of Zouakia. Furthermore, for the Lipschitz algebras and the algebra of absolutely continuous functions on , we characterize the closed sets for which the restriction algebras and are generated by their idempotents.
Soit l’algèbre de Beurling à poids sur le cercle unité et, pour un ensemble fermé , soit . Nous montrons que, pour , il existe un ensemble fermé de mesure nulle tel que l’algèbre quotient n’est pas engendrée par ses idempotents, contrastant par là avec un résultat de Zouakia. De plus, pour les algèbres de Lipschitz et l’algèbre des fonctions absolument continues sur , nous caractérisons les ensembles fermés tels que les algèbres restrictions et 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}, 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/} }
TY - JOUR AU - Pedersen, Thomas Vils TI - Idempotents in quotients and restrictions of Banach algebras of functions JO - Annales de l'Institut Fourier PY - 1996 SP - 1095 EP - 1124 VL - 46 IS - 4 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.1542/ DO - 10.5802/aif.1542 LA - en ID - AIF_1996__46_4_1095_0 ER -
%0 Journal Article %A Pedersen, Thomas Vils %T Idempotents in quotients and restrictions of Banach algebras of functions %J Annales de l'Institut Fourier %D 1996 %P 1095-1124 %V 46 %N 4 %I Association des Annales de l’institut Fourier %U http://www.numdam.org/articles/10.5802/aif.1542/ %R 10.5802/aif.1542 %G en %F 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/articles/10.5802/aif.1542/
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