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

TL;DR

This paper investigates the structure of Banach spaces where points of weak* to weak continuity are dense and form a G_delta set, analyzing their behavior in higher duals and specific algebraic contexts.

Contribution

It characterizes classes of Banach spaces with dense weak* to weak continuity points and explores their properties in higher duals and von Neumann algebras.

Findings

Points of weak* to weak continuity can be dense and G_delta in certain Banach spaces.

In von Neumann algebras with preduals having the Radon-Nikodym property, such points do not exist.

The property varies significantly in higher duals of Banach spaces.

Abstract

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 $\mathcal{A}$.