  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.
