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Abstract

We give sufficient conditions on Banach spaces $E$ and $F$ so that their projective tensor product $E\otimes _\pi F$ and the duals of their projective and injective tensor products do not have the Dunford–Pettis property. We prove that if $E^*$ does not have the Schur property, $F$ is infinite-dimensional, and every operator $T:E^*\to F^{**}$ is completely continuous, then $(E\otimes _\epsilon F)^*$ does not have the DPP. We also prove that if $E^*$ does not have the Schur property, $F$ is infinite-dimensional, and every operator $T: F^{**} \to E^*$ is completely continuous, then $(E\otimes _\pi F)^*\simeq L(E,F^*)$ does not have the DPP.