Locally almost square Banach lattices

TL;DR

This paper investigates the relationship between local almost squareness and weak almost squareness in Banach lattices, introducing a positive variant that implies the diameter two property and provides new examples of such spaces.

Contribution

It introduces a positive variant of local almost squareness in Banach lattices and shows it implies the diameter two property, offering new examples of these spaces.

Findings

Positive local almost squareness implies diameter two property.

The positive variant helps generate new examples of diameter two spaces.

The study clarifies the relationship between local and weak almost squareness in Banach lattices.

Abstract

A Banach space is locally almost square if, for every $y$ in its unit sphere, there exists a sequence $(x_n)$ in its unit sphere such that $\lim\|y\pm x_n\|=1$. A Banach space is weakly almost square if, in addition, we require the sequence $(x_n)$ to be weakly null. It is known that these two properties are distinct, so we aim to investigate if local almost squareness implies a weaker version of the latter property by replacing the sequence with a net. In order to achieve this result, we restrict ourselves to Banach lattices and introduce a strengthening of local almost squareness by requiring that the sequence is in the positive cone of the lattice. As an application of such characterization, we prove that this positive variant of local almost squareness implies that every relatively weakly open set in the unit ball has diameter 2, that is, the Banach space has the so called diameter two property. This in particular allows us also to generate new examples of Banach spaces enjoying the diameter two property.