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–Nikodým property, then there is no point of weak \(^*\) -weak continuity on the unit ball of \(\mathcal {A}\) .
