A characterization of the weak topology in the unit ball of purely atomic L1 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 L1 preduals with a weak stable unit ball agree with those L1 preduals which are purely atomic, that is preduals of ℓ1(Γ) for some set Γ, getting in this way a complete geometrical characterization of purely atomic preduals of L1, which answers a setting problem. As a consequence, we prove the equivalence for L1 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 C0(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 C0(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.