Éléments de Cohen et fonctions extérieures de l'algèbre du disque. II
Rajoelina, Michel M.
Annales de l'Institut Fourier, Tome 39 (1989), p. 1061-1072 / Harvested from Numdam

Nous étudions ici les éléments de Cohen de K K désigne l’ensemble des éléments de l’algèbre du disque nuls sur K quand K est un ensemble de mesure nulle du cercle. Nous montrons qu’une fonction f K est un élément de Cohen de K si et seulement si f est extérieure et s’annule exactement sur K.

The notion of a “Cohen element” was introduced for commutative separable Banach algebra 𝒜 with bounded approximate identity as a tool to construct discontinuous homomorphism from 𝒞(X). Such elements generate in particular dense principal ideals in 𝒜.

We study here these elements in the case of the algebra K of elements of the disc algebra vanishing on a closed negligible subset K of the unit circle. We show that the set of Cohen elements of K is exactly the set of elements of K which generate a dense principal ideal of K . In other terms a function f belonging to K is a Cohen element if and only if f is an outer function vanishing on K and only on K.

Publié le : 1989-01-01
DOI : https://doi.org/10.5802/aif.1199
@article{AIF_1989__39_4_1061_0,
     author = {Rajoelina, Michel M.},
     title = {\'El\'ements de Cohen et fonctions ext\'erieures de l'alg\`ebre du disque. II},
     journal = {Annales de l'Institut Fourier},
     volume = {39},
     year = {1989},
     pages = {1061-1072},
     doi = {10.5802/aif.1199},
     mrnumber = {92a:46060},
     language = {fr},
     url = {http://dml.mathdoc.fr/item/AIF_1989__39_4_1061_0}
}
Rajoelina, Michel M. Éléments de Cohen et fonctions extérieures de l'algèbre du disque. II. Annales de l'Institut Fourier, Tome 39 (1989) pp. 1061-1072. doi : 10.5802/aif.1199. http://gdmltest.u-ga.fr/item/AIF_1989__39_4_1061_0/

[1] N. U. Arakelian, Approximation complexe et propriétés des fonctions analytiques, Actes Congrès intern. Math., 1970, t. 2, 595-600. | Zbl 0234.30029

[2] H. G. Dales, A discontinuous homomorphism from C (X), Am. J. Math., 101 (1979), 647-734. | MR 81g:46066 | Zbl 0417.46054

[3] J. Esterle, Injection de semi-groupes divisibles dans des algèbres de convolution et construction d'homomorphismes discontinus de C (K), Proc. Lond. Math. Soc., 36 (1978), 59-85. | MR 58 #2300 | Zbl 0411.46039

[4] J. Esterle, Universal properties of some commutative radical Banach algebras, J. Reine Ang. Math., 321 (1981), 1-24. | MR 82i:46078 | Zbl 0438.46037

[5] J. Esterle, Elements for a classification of commutative radical Banach algebras, Lect. Notes Math., 975, pp. 4-65, Berlin, Heidelberg, New-York, Springer, 1983. | MR 84h:46064 | Zbl 0569.46030

[6] K. Hoffman, Banach spaces of analytic functions, Prentice-Hall, Englewwood cliffs, 1962. | MR 24 #A2844 | Zbl 0117.34001

[7] M. M. Rajoelina, Éléments de Cohen et fonctions extérieures de l'algèbre du disque, Bull. Sci. Math. (à paraître). | Zbl 0752.46032

[8] S. Scheinberg, Uniform approximation by functions analytic on a Riemann surface, Ann. of Math., 108 (1978), 257-298. | MR 58 #17111 | Zbl 0423.30035

[9] A. M. Sinclair, Cohen elements in Banach algebras, Proc. R. Edinb., Sect. A, 84 (1979), 55-70. | MR 81c:46044 | Zbl 0425.46038

[10] F. Zouakia, The theory of Cohen elements, Lect. Notes Math., 975, pp. 163-178, Berlin, Heidelberg, New York, Springer, 1983. | MR 84f:46069 | Zbl 0504.46037