Projective tensor products where every element is norm-attaining

In this paper we analyse when every element of X_ Y attains its projective norm. We prove that this is the case if X is the dual of a subspace of a predual of an ₁ (I) space and Y is 1-complemented in its bidual under approximation properties assumptions. This result allows us to provide some new examples where X is a Lipschitz-free space. We also prove that the set of norm-attaining elements is dense in X_ Y if, for instance, X=L₁ () and Y is any Banach space, or if X has the metric -property and Y is a dual space with the RNP.