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

Abstract

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.