Moebius-invariant algebras in balls

Rudin, Walter

Annales de l'Institut Fourier, Tome 33 (1983) no. 2, pp. 19-41.

Abstract (VO)

It is proved that the Fréchet algebra $C\left(B\right)$ has exactly three closed subalgebras $Y$ which contain nonconstant functions and which are invariant, in the sense that $f\circ \Psi \in Y$ whenever $f\in Y$ and $\Psi$ is a biholomorphic map of the open unit ball $B$ of ${\mathbf{C}}^n$ onto $B$. One of these consists of the holomorphic functions in $B$, the second consists of those whose complex conjugates are holomorphic, and the third is $C\left(B\right)$.

Résumé

On démontre que dans l’algèbre de Fréchet $C\left(B\right)$ il y a exactement trois sous-algèbres $Y$ qui sont fermées, qui contiennent des fonctions non constantes, et qui sont invariantes dans le sens suivant : $f\circ \Psi \in Y$ lorsque $f\in Y$ et $\Psi$ est une application biholomorphe de la boule unité ouverte $B$ de ${\mathbf{C}}^n$ sur $B$. Ce sont (i) l’algèbre des fonctions holomorphes dans $B$, (ii) l’algèbre des fonctions $f$ dont les conjuguées $\overline{f}$ sont holomorphes, (iii) $C\left(B\right)$.

Détail

DOI : 10.5802/aif.914

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