Several Remarks on Norm Attainment 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 (circle times) over cap Y-pi 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 (l(p)(I)(circle times) over cap (pi)l(q)(J))(*) has the w(*)-Kadec-Klee property for every non-empty sets I,J and every 2<p,q<infinity, obtaining in particular that the norm of the space l(p)(I)(circle times) over cap (pi)l(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 (circle times) over cap Y-pi is dense but whose complement is dense too.