A characterization of the weak topology in the unit ball of purely atomic $L_1$ preduals

Abstract

We study Banach spaces with a weak stable unit ball, that is Banach spaces where every convex combination of relatively weakly open subsets in its unit ball is again a relatively weakly open subset in its unit ball. It is proved that the class of $L_1$ preduals with a weak stable unit ball agree with those $L_1$ preduals which are purely atomic, that is preduals of $\ell_1(\Gamma)$ for some set $\Gamma$, getting in this way a complete geometrical characterization of purely atomic preduals of $L_1$, which answers a setting problem. As a consequence, we prove the equivalence for $L_1$ preduals of different properties previously studied by other authors, in terms of slices around weak stability. Also we get the weak stability of the unit ball of $C_0(K,X)$ whenever $K$ is a Hausdorff and scattered locally compact space and $X$ has a norm stable and weak stable unit ball, which gives the weak stability of the unit ball in $C_0(K,X)$ for finite-dimensional $X$ with a stable unit ball and $K$ as above. Finally we prove that Banach spaces with a weak stable unit ball satisfy a very strong new version of diameter two property.