Non Dentable Sets in Banach Spaces with Separable Dual

A non RNP Banach space E is constructed such that E∗ is separable and RNP is equivalent to PCP on the subsets of E. The problem of the equivalence of the Radon-Nikodym Property (RNP) and the Krein Milman Property (KMP) remains open for Banach spaces as well as for closed convex sets. A step forward has been made by Schachermayer’s Theorem [S]. That result states that the two properties are equivalent on strongly regular sets. Rosenthal, [R], has shown that every non-RNP strongly regular closed convex set contains a non-dentable subset on which the norm and weak topologies coincide. In a previous paper ([A-D]) we proved that every non RNP closed convex contains a subset with a martigale coordination. Furthermore we established the Pαl-representation for several cases. The remaining open case in the equivalence of RNP and KMP is that of B-spaces or closed convex sets where RNP is equivalent to PCP in their subsets. Typical example for a such structure are the subsets of L(0, 1). H. Rosenthal raised the question if this could occur when the dual of the space is separable. W. James ([J2]) also posed a similar problem. The aim of the present paper is to give an example of a Banach space E with separable dual failing RNP, and RNP is equivalent to PCP on its subsets. As consequence we get that E does not contain co(N) isomorphically and hence it does not embed into a Banach space with an unconditional skipped F.D.D. On the other hand E semiembeds into a Banach space with an unconditional basis. The last property allows us to conclude that every closed convex non-RNP subset of E contains a closed non-dentable set with a Pαl-representation. We recall that a closed set K has a Pαl-representation if there is an affine, onto, one to one continuous map from the atomless probability measures on [0,1] to the set K. In particular RNP is equivalent to KMP on the subsets of E. The space E is realized by applying the Davis-Figiel-JohnsonPelczynski factorization method to a convex symmetric set W of a Banach space Eu constructed in this paper. Finally as a consequence of the methods used in the proofs of the example we obtain that every separable B-space X such that X/X is isomorphic to l(Γ) has RNP. We thank H. Rosenthal and T. Odell for some useful discussions related to the problem studied in the present paper. We also thank the Department of Mathematics of Oklahoma State University for its technical support. Typeset by AMS-TEX 2 SPIROS A. ARGYROS AND IRENE DELIYANNI (HERAKLEION CRETE) We start with some definitions, notations and results necessary for our constructions. A closed convex bounded set K is said to be δ-non dentable, δ > 0, if every slice of K has diameter greater than δ. A closed convex set has RNP if it contains no δ-non dentable set. A closed K subset of a B-space has the P.C.P. if for every subset L of K and for all ε > 0 there exists a relatively weakly open neibhd of L with diameter less than ε. It is well known that RNP implies P.C.P, but the converse fails [B-R]. In the sequel D denotes the dyadic tree namely the set of all finite sequences of the for a = {0, ε1, ..., εn} with εi = 0 or 1. For a in D the length of α is denoted by |a|. A natural order is induced on D, that is a ≺ β if the sequence a is an initial segment of the sequence β. Two elements a, β of D are called incomparable if they are imcomparable in the above defined order. We notice, for later use, that each a in D determines a unique basic clopen subset Va in Cantor’s group {0, 1} N and a, β are imcomparable if Va ∩ Vβ = ∅. A basic ingredient in the definition of the space E is Tsirelson’s norm as it is defined in [F-J]. We recall that the norm of this space satisfies the following implicit fixed point property. For x = m ∑