Homogeneous algebras on the circle. II. Multipliers, Ditkin conditions

Bennett, Colin; Gilbert, John E.

Annales de l'Institut Fourier, Tome 22 (1972) no. 3, pp. 21-50.

Ce travail considère les sous-algèbres de Lipschitz $\Lambda (\alpha ,p,𝒜)$ d’une algèbre homogène sur le cercle. La théorie des espaces d’interpolation est utilisée pour dériver des estimations pour les normes des multiplicateurs sur des idéaux primaires fermés de $\Lambda (\alpha ,p;𝒜)$, $\alpha \ne \left[\alpha \right]$. D’après ces estimations on déduit facilement la condition de Ditkin et la condition analytique de Ditkin pour $\Lambda (\alpha ,p,𝒜)$. De cette façon la théorie familière des algèbres (régulières) de Banach qui satisfont à la condition de Ditkin et aussi la théorie développée dans la Note I de cette série qui exige que la condition analytique de Ditkin s’applique à $\Lambda (\alpha ,p,𝒜)$.

On présente des exemples qui démontrent que beaucoup d’algèbres de Banach considérées précédemment séparément peuvent être introduites aussi bien qu’analysées dans ce cadre de la théorie des espaces d’interpolation.

This paper considers the Lipschitz subalgebras $\Lambda (\alpha ,p,𝒜)$ of a homogeneous algebra on the circle. Interpolation space theory is used to derive estimates for the multiplier norm on closed primary ideals in $\Lambda (\alpha ,p;𝒜)$, $\alpha \ne \left[\alpha \right]$. From these estimates the Ditkin and Analytic Ditkin conditions for $\Lambda (\alpha ,p;𝒜)$ follow easily. Thus the well-known theory of (regular) Banach algebras satisfying the Ditkin condition applies to $\Lambda (\alpha ;,p;𝒜)$ as does the theory developed in part I of this series which requires the Analytic Ditkin condition.

Examples are discussed showing that many of the Banach algebras on the circle considered previously in isolation can be both generated and describes within this framework of interpolation space theory.

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