A characterization of the weak topology in the unit ball of purely atomic L1 preduals

We study Banach spaces with a weak stable unit ball, that is, where every convex combination of relatively weakly open subsets in its ball is again subset ball. It proved the class L1 preduals agree those which are purely atomic, ℓ1(Γ) for some set Γ, getting this way complete geometrical characterization atomic L1, answers setting problem. As consequence, we prove equivalence different properties previously studied by other authors, terms slices around stability. Also get stability C0(K,X) whenever K Hausdorff and scattered locally compact space X has norm This gives finite-dimensional X. Finally satisfy very strong new version diameter two property.