Ensembles de Rosenthal et propriété de Radon-Nikodym relative

Daher, Mohammad

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 18 (2009) no. 3, pp. 599-610.

Soient $G$ un groupe abélien compact métrisable, $\Gamma$ son groupe dual et $\Lambda \subset \Gamma$ un ensemble de Rosenthal. Nous montrons que $L_{\Lambda }^{\infty }(G,Y^{*})=C_{\Lambda }(G,Y^{*}),$ lorsque $Y^{*}$ est un espace de Banach ayant la propriété de Radon-Nikodym et $C_{\Lambda }(G,Y^{*})$ est faiblement séquentiellement complet. Nous en déduisons une condition suffisante pour que le produit de deux ensembles de Rosenthal en soit encore un pour le groupe produit. Ensuite nous introduisons la propriété de Radon-Nikodym relative $RN$-$\Lambda$, une généralisation de la propriété de Radon-Nikodym analytique. Nous montrons que si $L^1\left(G\right)/L_{{\Lambda }^c}^1\left(G\right)$ a la propriété $RN$-$\Lambda$, alors $\Lambda$ est fini. Cela nous permet de retrouver très simplement le fait que $L^1\left(𝕋\right)/H^1\left(𝕋\right)$ n’a pas la propriété de Radon-Nikodym analytique

Let $G$ be a metrizable compact abelian group, $\Gamma$ its dual group and let $\Lambda \subset \Gamma$ be a Rosenthal set. We show that $L_{\Lambda }^{\infty }(G,Y^{*})=C_{\Lambda }(G,Y^{*})$ whenever $Y^{*}$ is a Banach space with Radon-Nikodym property and $C_{\Lambda }(G,Y^{*})$ is weakly sequentially complete. We deduce a condition implying that the product of two Rosenthal sets is still a Rosenthal set in product group. Then we introduce the relative Radon-Nikodym property $RN$-$\Lambda$, which generalizes the analytic Radon-Nikodym property. We prove that $RN$-$\Lambda$ property for $L^1\left(G\right)/L_{{\Lambda }^c}^1\left(G\right)$ implies that $\Lambda$ is finite. This gives a new and easy proof that $L^1\left(𝕋\right)/H^1\left(𝕋\right)$ does not possess the analytic Radon-Nikodym property.

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