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 ℓ 1 ( Γ ) for some set Γ, 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. This gives a characterization of weak stability of the unit ball in C 0 ( K , X ) for finite-dimensional X . Finally we prove that Banach spaces with a weak stable unit ball satisfy a very strong new version of diameter two property.
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