Daugavet points in projective tensor products

TL;DR

This paper investigates conditions under which elements in projective tensor products are Daugavet points, revealing connections to isometries and providing methods to construct such points, especially in spaces like $C(K)$.

Contribution

It establishes links between the Daugavet property in tensor products and isometries, and offers new methods to identify Daugavet points in various Banach spaces.

Findings

Daugavet property implies many isometries from Y into X*

Methods for constructing non-trivial Daugavet points

Daugavet points are weakly dense in certain tensor products with C(K) spaces

Abstract

In this paper, we are interested in studying when an element $z$ in the projective tensor product $X \widehat{\otimes}_\pi Y$ turns out to be a Daugavet point. We prove first that, under some hypothesis, the assumption of $X \widehat{\otimes}_\pi Y$ having the Daugavet property implies the existence of a great amount of isometries from $Y$ into $X^*$. Having this in mind, we provide methods for constructing non-trivial Daugavet points in $X \widehat{\otimes}_\pi Y$. We show that $C(K)$-spaces are examples of Banach spaces such that the set of the Daugavet points in $C(K) \widehat{\otimes}_\pi Y$ is weakly dense when $Y$ is a subspace of $C(K)^*$. Finally, we present some natural results on when an elementary tensor $x \otimes y$ is a Daugavet point.