The Dunford–Pettis property on tensor products

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 (c0 &[otimes ]circ;πc0)** fails the DPP. Since (c0 &[otimes ]circ;πc0)* 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 [Lscr ]1-spaces, then E &[otimes ]circ;ε has the DPP if and only if both E and F have the Schur property. Other results and examples are given.