New examples of strongly subdifferentiable projective tensor products

TL;DR

This paper investigates conditions under which the projective tensor product of certain Banach spaces exhibits strongly subdifferentiable (SSD) norms, providing new examples and characterizations in the context of functional analysis.

Contribution

It introduces new classes of Banach space tensor products with SSD norms and characterizes SSD elements via a strengthened local Bollobás property.

Findings

SSD norm of $X\widehat{\otimes}_\pi Y$ when $X=\ell_p(I)$, $p>2$, and $Y$ has specific convexity properties.

SSD norm when $X=c_0(I)$ and $Y^*$ is uniformly convex finite-dimensional.

Characterization of SSD elements that attain the projective norm in tensor products.

Abstract

We prove that the norm of $X\widehat{\otimes}_\pi Y$ is SSD if either $X=\ell_p(I)$ for $p>2$ and $Y$ is a finite-dimensional Banach space such that the modulus of convexity is of power type $q<p$ (e.g. if $Y^*$ is a subspace of $L_q$) or if $X=c_0(I)$ and $Y^*$ is any uniformly convex finite-dimensional Banach space. We also provide a characterisation of SSD elements of a projective tensor product which attain its projective norm in terms of a strengthening of the a local Bollob\'as property for bilinear mappings.