Dunford - Pettis Operators 3

If all bounded linear operators from L 1 into a Banach space X are Dunford-Pettis (i.e. carry weakly convergent sequences onto norm convergent sequences), then we say that X has the complete continuity property (CCP). The CCP is a weakening of the Radon-Nikod ym property (RNP). Basic results of Bourgain and Talagrand began to suggest the possibility that the CCP, like the RNP, can be realized as an internal geometric property of Banach spaces; the purpose of this paper is to provide such a realization. We begin by showing that X has the CCP if and only if every bounded subset of X is Bocce dentable, or equivalently, every bounded subset of X is weak-norm-one dentable (Section 2). This internal geometric description leads to another; namely, X has the CCP if and only if no bounded separated-trees grow in X, or equivalently, no bounded-Rademacher trees grow in X (Section 3). ((; ;) refers to the Lebesgue measure space on 0; 1], + to the sets in with positive measure, and L 1 to L 1 ((; ;). All notation and terminology, not otherwise explained, are as in DU]. For clarity, known results are presented as facts while new results are presented as theorems, lemmas, and observations. The following fact provides several equivalent formulations of the CCP.