Daugavet Points in Projective Tensor Products

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.

Daugavet- and Delta-points in spaces of Lipschitz functions

“…These concepts were introduced as local versions of the Daugavet property and the diametral local diameter two property, respectively (see [1], [15]). The properties of these points have been studied in various papers (see, e.g., [2], [6], [10], [12], [13]).…”

Characterizations of Daugavet points and delta-points in Lipschitz-free spaces

“…These recently introduced concepts opened a new direction in the study of the well-known Daugavet property and diameter-2 properties (see, e.g., [2], [6], [10], [12]). In fact, a Banach space has the Daugavet property (respectively, diametral local diameter-2 property) if and only if all unit sphere elements are Daugavet points (respectively, ∆-points) (see [1], [14]).…”