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Cohen elements and exterior functions of the disk algebra. (Éléments de Cohen et fonctions extérieures de l’algèbre du disque.) (French) Zbl 0752.46032
Let $$A$$ be a commutative algebra. An element $$x\in A$$ is called a Cohen element if there exists a sequence $$(a_ n)$$ in the algebra $$A\oplus\mathbb{C}\cdot e$$ obtained by adjoining an identity to $$A$$ such that
(1) $$x=\lim_{n\to\infty}\exp a_ n$$.
(2) For every $$j\in\mathbb{N}$$, $$(\exp({1\over j}a_ n))_{n=1}^ \infty$$ converges to an element of $$A$$ denoted by $$x^{1/j}$$.
(3) $$\sup_{n\in\mathbb{N}}\| x^{1/j}\exp(-{1\over j}a_ n)\|<\infty$$ for every $$j\in\mathbb{N}$$.
(4) $$(ax\exp(-a_ n))_{n=1}^ \infty$$ converges to $$a$$ for every $$a\in A$$.
It is clear that every Cohen element generates a dense principal ideal in $$A$$. Now let $$A(\mathbb{D})$$ denote the disk algebra.
In the paper under review the author shows that in case $$A=M_ 1:=\{f\in A(\mathbb{D}),\;f(1)=0\}$$ the converse holds, too. In particular, the Cohen elements of $$M_ 1$$ are exactly the outer functions vanishing at $$z=1$$ and nowhere else. In a subsequent paper [Ann. Inst. Fourier (Grenoble) 39, 1061-1072 (1989)] the author could generalize this result to the algebra $$M_ K=\{f\in A(\mathbb{D}),\;f\mid_ K\equiv 0\}$$, where $$K$$ is a compact subset of measure zero of the unit circle.

##### MSC:
 46J10 Banach algebras of continuous functions, function algebras