Intersection properties of balls in spaces of compact operators

Lima, Asvald

Annales de l'Institut Fourier, Tome 28 (1978) no. 3, pp. 35-65.

Cet article est une étude des rapports entre les propriétés d’intersection et l’existence des grandes faces pour les boules d’un espace de Banach. D’après un résultat classique de Hanner un espace de dimension finie a la propriété d’intersection 3.2 (la “p.i. 3.2”) si et seulement si deux faces disjointes quelconques sont contenues dans deux hyperplans parallèles. Nous donnons ici une démonstration pour le cas général. Nous prouvons aussi que l’espace $C(X,Y)$ des opérateurs compacts de $X$ dans $Y$ a la p.i. 3.2 si et seulement si $X$ et $Y$ ont la p.i. 3.2 et de plus ou bien $X$ ou bien $Y^{*}$ est isométrique à un espace $L_1\left(\mu \right)$.

We study the connection between intersection properties of balls and the existence of large faces of the unit ball in Banach spaces. Hanner’s result that a real space has the 3.2 intersection property if an only if disjoint faces of the unit ball are contained in parallel hyperplanes is extended to infinite dimensional spaces. It is shown that the space of compact operators from a space $X$ to a space $Y$ has the 3.2 intersection property if and only if $X$ and $Y$ have the 3.2 intersection property and either $X$ or $Y^{*}$ is isometric to an $L_1\left(\mu \right)$-space.

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