Abstract
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 Xwidehat{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)widehat{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)widehat{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 Xwidehat{otimes }_pi Y is dense but whose complement is dense too.
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