A study of weak$^*$-weak points of continuity in the unit ball of dual spaces

We study classes of Banach spaces where the points of weak$^*$-weak continuity for the identity mapping on the dual unit ball form a weak$^*$-dense and weak$^*$-$G_{\delta}$ set. We also discuss how this property behaves in higher duals of Banach spaces. We prove in particular that if $\mathcal{A}$ is a von Neumann algebra and its predual has the Radon--Nikod\'ym property, then there is no point of weak$^*$-weak continuity on the unit ball of $\mathcal{A}$.