On Hahn-Banach smoothness of $L_1$-preduals and related $w^*-w$ point of continuity of unit balls of dual spaces

Abstract: In this article, we intend to study possible $(U)$-embeddings of a Banach space $X$ into $X^{**}$. The canonical embedding of $X$ in $X^{**}$ which possesses $(U)$-embedding is of particular interest and such spaces are known as Hahn-Banach smooth spaces. Separable $L_1$-preduals are characterized which are Hahn-Banach smooth. It is derived that, when $S$ is a compact convex set where each point in $ext(S)$ is a limit point of $ext(S)$ then no subspace of $A(S)$ retains property-$(wU)$ in $A(S)^{**}$. Moreover, if $X$ is an $L_1$-predual where $I:(B_{X^},w^)\rightarrow (B_{X^*},w)$ is continuous on $ext (B_{X^*})$ then $X$ is Hahn-Banach smooth, is observed. This means that not all finitely supported elements in $B_{\ell_1}$ can be points of continuity of $I:(B_{\ell_1},w^*(c))\rightarrow (B_{\ell_1},w)$, which is incorrectly stated in \cite{DMR}. Throughout this article this fact is established in a few ways. It is shown that if $L_1(μ)$ possesses a predual that is weakly Hahn-Banach smooth, then $μ$ must have a specific characteristic.