$L$-orthogonality, octahedrality and Daugavet property in Banach spaces

TL;DR

This paper investigates the relationships between $L$-orthogonality, octahedrality, and the Daugavet property in nonseparable Banach spaces, revealing new conditions and limitations, especially concerning spaces with density character $ ext{ extomega}_1$ and under the continuum hypothesis.

Contribution

It demonstrates that octahedrality does not imply $L$-orthogonality in nonseparable spaces and characterizes the Daugavet property via $L$-orthogonality density under certain set-theoretic assumptions.

Findings

Octahedrality does not imply $L$-orthogonality in nonseparable spaces.

Density character $ ext{ extomega}_1$ spaces with almost Daugavet property have abundant $L$-orthogonal vectors.

Under CH, the characterization of the Daugavet property via $L$-orthogonality density fails for larger spaces.

Abstract

In contrast with the separable case, we prove that the existence of almost $L$-orthogonal vectors in a nonseparable Banach space $X$ (octahedrality) does not imply the existence of nonzero vectors in $X^{**}$ being $L$-orthogonal to $X$, which shows that the answer to an environment question in [9] is negative. Furthermore, we prove that the abundance of almost $L$-orthogonal vectors in a Banach space $X$ (almost Daugavet property) whose density character is $\omega_1$ implies the abundance of nonzero vectors in $X^{**}$ being $L$-orthogonal to $X$. In fact, we get that a Banach space $X$ whose density character is $\omega_1$ verifies the Daugavet property if, and only if, the set of vectors in $X^{**}$ being $L$-orthogonal to $X$ is weak-star dense in $X^{**}$. We also prove that, under CH, the previous characterisation is false for Banach spaces with larger density character. Finally, some consequences on Daugavet property in the setting of $L$-embedded spaces are obtained.