A characterization of the weak topology in the unit ball of purely atomic \(L_1\) preduals

Abstract

We study weak stable Banach spaces, 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 weak stable \(L_1\) preduals 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 an environment 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. Finally we prove that every infinite-dimensional purely atomic \(L_1\) predual satisfies the strongest diameter two property and get consequences about the behavior of the set of extreme points in the unit ball of these spaces with respect the weak topology.