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Several remarks on norm attaining in tensor product spaces

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 $X\hat{\otimes}_πY$ 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 $(\ell_p(I)\hat{\otimes}_π\ell_q(J))^*$ 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 $\ell_p(I)\hat{\otimes}_π\ell_q(J)$ 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.