Projective tensor products where every element is norm-attaining

Abstract

In this paper we analyse when every element of $X{\hat{\otimes }}_{\ddot{I}}€Y$ 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 $X{\hat{\otimes }}_{\ddot{I}}€Y$ if, for instance, $X=L_1(\hat{I}¼)$ and $Y$ is any Banach space, or if $X$ has the metric $\ddot{I}€$-property and $Y$ is a dual space with the RNP.