×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite