The Alternative Dunford-pettis Property on Projective Tensor Products

In 1953 A. Grothendieck introduced the property known as Dunford-Pettis property [18]. A Banach space X has the Dunford-Pettis property (DPP in the sequel) if whenever (xn) and (ρn) are weakly null sequences in X and X∗, respectively, we have ρn(xn) → 0. It is due to Grothendieck that every C(K )-space satisfies the DPP. Historically, were Dunford and Pettis who first proved that L1(μ) satisfies DPP. Since its introduction, the DPP has been intensively studied and developed in many classes of Banach spaces. In the last twenty years the problem of determine when the projective tensor product of two Banach spaces satisfies the Dunford-Pettis property has focussed the attention of several researchers. Since the DPP is inherited by complemented subspaces, it follows that X and Y satisfy the DPP whenever X⊗̂πY has this property. However, M. Talagrand showed in [26] that this necessary condition is not always sufficient by finding a Banach space X such that X∗ has the Schur property and X∗⊗̂π L1[0, 1] does not satisfy the DPP. In 1987, R. Ryan proved that X⊗̂πY satisfy the DPP and does not contain a subspace isomorphic to 1 if and only if X and Y have both properties.