  The aim of this note is to obtain results about when the norm of a projective
tensor product is strongly subdifferentiable. We prove that if
$X\hat{\otimes}_\pi Y$ is strongly subdifferentiable and either $X$ or $Y$ has
the metric approximation property then every bounded operator from $X$ to $Y^*$
is compact. We also prove that $(\ell_p(I)\hat{\otimes}_\pi \ell_q(J))^*$ has
the $w^*$-Kadec-Klee property for every non-empty sets $I,J$ and every
$2<p,q<\infty$, obtaining in particular that the norm of the space
$\ell_p(I)\hat{\otimes}_\pi \ell_q(J)$ is strongly subdifferentiable. This
extends several results of Dantas, Kim, Lee and Mazzitelli. We also find
examples of spaces $X$ and $Y$ for which the set of norm-attaining tensors in
$X\pten Y$ is dense but whose complement is dense too.
