Weak$^*$-weak points of continuity on the state spaces

Abstract

Let $X$ be a Banach space. For $x \in X$ with $\|x\| = 1$, we denote the state space by $S_x = \{x^* \in X^* : \|x^*\| = x^*(x) = 1\}.$ In this paper, we study weak$^*$-weak and weak$^*$-$\|\cdot\|$ points of continuity of the identity map on the state spaces in the space $\ell^p(X)$ for $1 < p < \infty$, where $X$ is a non-reflexive Banach space. We then use these results to characterize the weak and norm compactness of the state spaces of unit vectors in $\ell^p(X)$. In addition, we address an open problem concerning the characterization of weakly compact state spaces in the space of Bochner-integrable functions $L^1(\mu, X)$. We also provide a local solution to this problem without any additional assumptions on the Banach space $X$. Motivated by the work of S. Daptari, V. Montesinos, and T. S. S. R. K. Rao, we show that if the set of all weak$^*$-weak points of continuity of $L^1(\mu, X)_1^*$ is weakly dense in $L^1(\mu, X)_1^*$, then $X^*$ has the Radon-Nikod\'ym property (RNP).