Given $X, Y$ two Banach spaces, we can consider the projective tensor product $X\widehat{\otimes}_\pi Y$
which is used to linearize bilinear functions. In this talk 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.
This work has been carried out in collaboration with Luis C. García-Lirola (Univesity of
Zaragoza) and Abraham Rueda Zoca (University of Granada).