The Dunford-Pettis property on tensor products

Abstract

We show that, in some cases, the projective and the injective tensor products of two Banach spaces do not have the Dunford-Pettis property (DPP). As a consequence, we obtain that $(c_0{\hat{\otimes }}_{\pi }{c}_0{)}^{\ast \ast }$ fails the DPP. Since $(c_0{\hat{\otimes }}_{\pi }{c}_0{)}^{\ast }$ does enjoy it, this provides a new space with the DPP whose dual fails to have it. We also prove that, if $E$ and $F$ are ${\mathcal{L}}_1$-spaces, then $E{\hat{\otimes }}_{\epsilon }F$ has the DPP if and only if both $E$ and $F$ have the Schur property. Other results and examples are given.

9 pages