The Dunford-Pettis property on tensor products

We show that, in some cases, the projective and the injective tensor produc ts of two Banach spaces do not have the Dunford-Pettis property (DPP). As a consequence, we obtain that (c(o) (x) over cap (pi) c(o))** fails the DPP. Since (co (x) over cap (pi) c(o))* does enjoy it, this provides a new spac e with the DPP whose dual fails to have it. We also prove that, if E and F are L-1-spaces, then E (x) over cap (epsilon) F has the DPP if and only if both E and F have the Schur property. Other results and examples are given.