Dunford-Pettis Sets, V*-Sets, and Property (MB*)

In this paper, we study how some localized properties, e.g., Dunford-Pettis sets, and V*-sets can be used to study more global structure properties. We introduce a new class of Banach spaces called Banach spaces with property (MB*). We say that a space X has property (MB*) if every V*-set in X is a Dunford-Pettis set in X. We characterize those spaces which have property (MB*). It is shown that X has property (MB*) if and only if for every Banach space Y, every unconditionally converging adjoint operator T* from X to Y* is completely continuous. Also, we recall property (MB) for Banach spaces and then study relation between these two properties.