On Hahn-Banach smoothness of $L_1$-preduals and related $w^*-w$ point of continuity of unit balls of dual spaces
Sainik Karak, Tanmoy Paul, Ryotaro Tanaka
TL;DR
The paper investigates when $L_1$-preduals exhibit Hahn-Banach smoothness and how this relates to the $w^*-w$ continuity of the identity on dual unit balls. It develops a framework linking HB-smoothness to $w^*$-ANP-III properties, reflexivity for dual spaces, and the discreteness of extreme points of dual unit balls, then characterizes preduals of $oxed{\ell_1}$ via relative $w^*$-$w$ continuity, showing that certain $L_1$-preduals must embed into $C_0(K)$ with a discrete spectrum. The work corrects a claim in prior literature about finitely supported elements in $oxed{\ell_1}$ being points of continuity for a $w^*-w$ map and demonstrates that even in non-$M$-embedded settings, HB-smoothness can arise under strong topological constraints. It also derives measure-theoretic consequences, proving that a weakly HB predual forces the underlying measure to be atomic, thereby linking functional-analytic smoothness to concrete measure structure.
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.