A study of weak${}^{\ast }$-weak points of continuity in the unit ball of dual spaces

Abstract

We study classes of Banach spaces where the points of weak${}^{\ast }$-weak continuity for the identity mapping on the dual unit ball form a weak${}^{\ast }$-dense and weak${}^{\ast }$-$G_{\delta }$ 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${}^{\ast }$-weak continuity on the unit ball of $\mathcal{A}$.

To appear in Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas