The point of continuity property in Banach spaces not containing ℓ1

Abstract

We obtain a local characterization of the point of continuity property for bounded subsets in Banach spaces not containing basic sequences equivalent to the standard basis of l1 and, as a consequence, we deduce that, in Banach spaces with a separable dual, every closed, bounded, convex and nonempty subset failing the point of continuity property contains a further subset which can be seen inside the set of Borel regular probability measures on the Cantor set in a weak-star dense way. Also, we characterize in terms of trees the point of continuity property of Banach spaces not containing l1, by proving that a Banach space not containing l1 satis- fies the point of continuity property if, and only if, every seminormalized weakly null tree has a boundedly complete branch.