Diameter two properties in James spaces

TL;DR

This paper investigates diameter two properties in certain James spaces, showing some spaces have strong diameter two properties while their duals do not, and identifying hyperplanes with duals that do.

Contribution

It establishes the presence of strong diameter two properties in specific James spaces and their hyperplanes, and characterizes the dual norms as octahedral.

Findings

$JH$ and $JH_ty$ satisfy the strong diameter two property

Dual spaces of these Banach spaces fail diameter two properties

A hyperplane $M$ of $JH_ty$ has a dual with $w^*$-strong diameter two property

Abstract

We study the diameter two properties in the spaces $JH$, $JT_\infty$ and $JH_\infty$. We show that the topological dual space of the previous Banach spaces fails every diameter two property. However, we prove that $JH$ and $JH_{\infty}$ satisfy the strong diameter two property, and so the dual norm of these spaces is octahedral. Also we find a closed hyperplane $M$ of $JH_\infty$ whose topological dual space enjoys the $w^*$-strong diameter two property and also $M$ and $M^*$ have an octahedral norm.