Projective tensor products where every element is norm-attaining

Abstract: In this paper we analyse when every element of $X\widehat{\otimes}\pi Y$ attains its projective norm. We prove that this is the case if $X$ is the dual of a subspace of a predual of an $\ell_1(I)$ 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 $X\widehat{\otimes}\pi Y$ if, for instance, $X=L_1(\mu)$ and $Y$ is any Banach space, or if $X$ has the metric $\pi$-property and $Y$ is a dual space with the RNP.