Projective tensor products where every element is norm-attaining

TL;DR

This paper investigates conditions under which every element in the projective tensor product of two Banach spaces attains its norm, providing new examples involving Lipschitz-free spaces and establishing density results for norm-attaining elements.

Contribution

It characterizes when all elements in the projective tensor product attain their norm, especially for spaces related to $ ext{L}_1$ and Lipschitz-free spaces, and proves density of norm-attaining elements in certain cases.

Findings

All elements attain their norm if $X$ is a dual of a subspace of a predual of $ ext{L}_1(I)$ and $Y$ is 1-complemented in its bidual.

The set of norm-attaining elements is dense if $X= ext{L}_1( extmu)$ and $Y$ is any Banach space.

Density also holds if $X$ has the metric $ extpi$-property and $Y$ is a dual with the RNP.

Abstract

In this paper we analyse when every element of $X\widehat{\otimes}_\pi Y$ attains its projective norm. We prove that this is the case if $X$ is the dual of a subspace of a predual of an $\ell_1(I)$ space and $Y$ is $1$-complemented in its bidual under approximation properties assumptions. This result allows us to provide some new examples where $X$ is a Lipschitz-free space. We also prove that the set of norm-attaining elements is dense in $X\widehat{\otimes}_\pi Y$ if, for instance, $X=L_1(\mu)$ and $Y$ is any Banach space, or if $X$ has the metric $\pi$-property and $Y$ is a dual space with the RNP.