The Dunford-Pettis and the Kadec-Klee properties on tensor products of JB*-triples

Abstract

We recall that a Banach spaceX satisfies the Dunford-Pettis property (DPP) if every weakly compact operator T from X to another Banach space is completely continuous, i.e. T maps weakly Cauchy sequences into norm convergent sequences. The question if the projective tensor product of two Banach spaces having the DPP also has this property has focused the attention of several researcher. In 1983, M. Talagrand found a Banach space X such that X∗ has the Schur property and L1[0, 1]⊗πE does not satisfy the DPP (see [34]). Four years latter R. Ryan showed that X⊗πY satisfies the DPP and contains no copies of `1 whenever X and Y have both properties [33]. F. Bombal and I. Villanueva prove in [7] (see also [6, 20]) that, whenever K1 and K2 are two infinite compact Hausdorff spaces, then the projective tensor product of C(K1) and C(K2) has the DPP if and only if K1 and K2 are scattered, that is C(K1) and C(K2) contains no copies of `1. The interesting ideas developed by Bombal, Fernandez and Villanueva in [7, 6] are the inspiration and the starting point for the present paper. In the