Projective tensor products where every element is norm-attaining

In this paper we analyse when every element of X⊗_π 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⊗_π Y if, for instance, X=L₁(μ) and Y is any Banach space, or if X has the metric π-property and Y is a dual space with the RNP.