Weak${}^{\ast }$-weak points of continuity on the state spaces

Let $X$ be a Banach space. For $x\in X$ with $\mathrm{\parallel }x\mathrm{\parallel }=1$, we denote the state space by $S_x=\{x^{\ast }\in X^{\ast }:\mathrm{\parallel }{x}^{\ast }\mathrm{\parallel }=x^{\ast }(x)=1\}\mathrm{.}$ In this paper, we study weak${}^{\ast }$-weak and weak${}^{\ast }$-$\mathrm{\parallel }\cdot \mathrm{\parallel }$ points of continuity of the identity map on the state spaces in the space ${\mathrm{\ell }}^p(X)$ for $1<p<\mathrm{\infty }$, 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 ${\mathrm{\ell }}^p(X)$. 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${}^{\ast }$-weak points of continuity of $L^1(\mu ,X{)}_1^{\ast }$ is weakly dense in $L^1(\mu ,X{)}_1^{\ast }$, then $X^{\ast }$ has the Radon-Nikodým property (RNP).