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If all bounded linear operators from L1 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). 1 : Introduction Throughout this paper, X denotes an arbitrary Banach space, X the dual space of X, B(X) the closed unit ball of X, and S(X) the unit sphere of X. The triple ( ; ; ) refers to the Lebesgue measure space on [0; 1], + to the sets in with positive measure, and L1 to L1( ; ; ). 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. 1991 Mathematics Subject Classi cation. 46B20, 47B38, 46B22, 28B99.

Dentability, trees, and Dunford-Pettis operators onL1

Dentability, trees, and Dunford-Pettis operators onL₁