Spaces of type $H^{\infty }+C$

Rudin, Walter

Annales de l'Institut Fourier, Tome 25 (1975) no. 1, pp. 99-125.

On démontre un théorème facile concernant une condition suffisante pour que la somme de deux sous-espaces fermés d’un espace de Banach soit fermée. Ce théorème conduit à plusieurs résultats du type du théorème de Sarason, qui dit que $H^{\infty }+C$ est une sous-algèbre fermée de $L^{\infty }$. Dans ces résultats, le cercle est remplacé par d’autres groupes, et au lieu du disque unité on considère les polydisques et boules dans les espaces de plusieurs variables complexes. Les sommes des idéaux fermés dans une algèbre de Banach sont aussi étudiés.

A simple theorem is proved which states a sufficient condition for the sum ot two closed subspaces of a Banach space to be closed. This leads to several analogues of Sarason’s theorem which states that $H^{\infty }+C$ is a closed subalgebra of $L^{\infty }$. In these analogues, the unit circle is replaces by other groups, and the unit disc is replaced by polydiscs or by balls in spaces of several complex variables. Sums of closed ideals in Banach algebras are also studied.

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