Diameter two properties in James spaces

Abstract

We study the diameter two properties in the (James type) Banach spaces $JH$, $JT_\infty$ and $JH_\infty$. We show that the dual spaces of these three Banach spaces fail every diameter two property. Also, we prove that $JH$ and $JH_{\infty}$ satisfy the strong diameter two property, and so the dual norms of these spaces are octahedral. In addition, we find a closed hyperplane $M$ of $JH_\infty$ such that its dual space, $M^*$, satisfies the $w^*$-strong diameter two property. Finally, we get that the natural norms of $M$ and $M^*$ are octahedral.