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
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Nous étudions ici les éléments de Cohen de ℳ K où ℳ 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∈ℳ K est un élément de Cohen de ℳ 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 ℳ 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 ℳ K is exactly the set of elements of ℳ K which generate a dense principal ideal of ℳ K . In other terms a function f belonging to ℳ 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 
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     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/

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