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 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},
     pages = {1095--1124},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {46},
     number = {4},
     year = {1996},
     doi = {10.5802/aif.1542},
     zbl = {0853.46047},
     mrnumber = {98b:46070},
     language = {en},
     url = {https://aif.centre-mersenne.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  - https://aif.centre-mersenne.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 https://aif.centre-mersenne.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. https://aif.centre-mersenne.org/articles/10.5802/aif.1542/

[1] W.G. Bade and H.G. Dales, The Wedderburn Decomposition of Some Commutative Banach Algebras, J. Funct. Anal., 107 (1992), 105-121. | MR | Zbl

[2] J.J. Benedetto, Spectral Synthesis, Academic Press, New York-London-San Francisco, 1975. | Zbl

[3] F.F. Bonsall and J. Duncan, Complete Normed Algebras, Springer-Verlag, Berlin-Heidelberg-New York, 1970. | Zbl

[4] I.M. Gelfand, D.A. Raikov and G.E. Shilov, Commutative Normed Rings, Chelsea Publishing Company, Bronx, New York, 1964.

[5] L.I. Hedberg, The Stone-Weierstrass theorem in Lipschitz algebras, Ark. Mat., 8 (1969), 63-72. | MR | Zbl

[6] E. Hewitt and K. Stromberg, Real and Abstract Analysis, Springer-Verlag, Berlin-Heidelberg-New York, 1965. | Zbl

[7] J.-P. Kahane, Séries de Fourier absolument convergentes, Springer-Verlag, Berlin-Heidelberg-New York, 1970. | MR | Zbl

[8] J.-P. Kahane and R. Salem, Ensembles parfaits et séries trigonométriques, Hermann, Paris, 1963. | MR | Zbl

[9] Y. Katznelson, An Introduction to Harmonic Analysis, John Wiley & Sons, New York, 1968. | MR | Zbl

[10] P. Malliavin, Impossibilité de la synthèse spectrale sur les groupes abeliens non compacts, Publ. Math. Inst. Hautes Etudes Sci., 2 (1959), 85-92. | Numdam | MR | Zbl

[11] H. Mirkil, The Work of Silov on Commutative Semi-simple Banach Algebras, volume 20 of Notas de Matemática. Instituto de Matemática Pura e Aplicada, Rio de Janeiro, 1959. | MR | Zbl

[12] H. Mirkil, Continuous translation of Hölder and Lipschitz functions, Can. J. Math., 12 (1960), 674-685. | MR | Zbl

[13] T.V. Pedersen, Banach Algebras of Functions on the Circle and the Disc, Ph. D. Dissertation, University of Cambridge, October 1994.

[14] C.E. Rickart, General Theory of Banach Algebras, D. Van Nostrand Company, Princeton, N.J., 1960. | MR | Zbl

[15] W. Rudin, Functional Analysis, McGraw-Hill Book Company, New York, 1973. | MR | Zbl

[16] D.R. Sherbert, The structure of ideals and point derivations in Banach algebras of Lipschitz functions, Trans. Amer. Math. Soc., 111 (1964), 240-272. | MR | Zbl

[17] G.E. Shilov, Homogeneous rings of functions, Amer. Math. Soc. Transl., 92, 1953, Reprinted in Amer. Math. Soc. Transl. (1), 8 (1962), 392-455. | Zbl

[18] F. Zouakia, Idéaux fermés de A+ et L1(ℝ+) et propriétés asymptotiques des contractions et des semigroupes contractants, Thèse pour le grade de Docteur d'Etat des Sciences, Université de Bordeaux I, 1990.

[19] A. Zygmund, Trigonometric Series, volume 1, Cambridge University Press, second edition, 1959. | Zbl

Cited by Sources: