The alternative Dunford-Pettis property on projective tensor products

Abstract

A Banach space X has the Dunford–Pettis property (DPP) if and only if whenever (xn) and (pn) are weakly null sequences in X and X*, respectively, we have pn(xn)→ 0. Freedman introduced a stricly weaker version of the DPP called the alternative Dunford–Pettis property (DP1). A Banach space X has the DP1 if whenever xn ! x weakly in X, with kxnk = kxk, and (xn) is weakly null in X*, we have that xn(xn)→ 0. The authors study the DP1 on projective tensor products of C*-algebras and JB*-triples. Their main result, Theorem 3.5, states that if X and Y are Banach spaces such that X contains an isometric copy of c0 and Y contains an isometric copy of C[0, 1], then Xˆ_Y , the projective tensor product of X and Y , does not have the DP1. As a corollary, they get that if X and Y are JB*-triples such that X is not reflexive and Y contains `1, then Xˆ_Y does not have the DP1. Furthermore, if A and B are infinite-dimensional C*-algebras, then Aˆ_B has the DPP if and only if Aˆ_B has the DP1 if and only if both A and B have the DPP and do not contain `1.