Éléments de Cohen et fonctions extérieures de l'algèbre du disque. II

Rajoelina, Michel M.

Annales de l'Institut Fourier, Tome 39 (1989) no. 4, pp. 1061-1072.

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 $f\in {ℳ}_K$ 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 $𝒞\left(X\right)$. 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 $K$.

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