Diameter two properties in James spaces

Abstract

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

arXiv comment19 pages