Uniqueness of Hahn-Banach extension and related norm-$1$ projections in dual spaces

TL;DR

This paper investigates the properties of subspaces in Banach spaces related to the uniqueness of Hahn-Banach extensions, their stability under tensor products, and characterizations in specific spaces like $c_0$ and $L_p$.

Contribution

It introduces and analyzes properties-$U$ and-$SU$ for subspaces, establishing their stability under tensor products and quotients, and characterizes hyperplanes with property-$SU$ in $c_0$.

Findings

Properties-$U$ and-$SU$ are stable under injective tensor products.

$L_p( u,Y)$ has property-$U$ (or-$SU$) in $L_p( u,X)$ iff $Y$ has it in $X$ when $X^*$ has Radon-Nikodym Property.

A smooth Banach space of dimension >3 is a Hilbert space iff certain subspace sum properties hold.

Abstract

In this paper we study two properties viz. property-$U$ and property-$SU$ of a subspace $Y$ of a Banach space which correspond to the uniqueness of the Hahn-Banach extension of each linear functional in $Y^*$ and in addition to that this association forms a linear operator of norm-1 from $Y^*$ to $X^*$. It is proved that, under certain geometric assumptions on $X, Y, Z$ these properties are stable with respect to the injective tensor product; $Y$ has property-$U$ ($SU$) in $Z$ if and only if $X\otimes_\e^\vee Y$ has property-$U$ ($SU$) in $X\otimes_\e^\vee Z$. We prove that when $X^*$ has the Radon-Nikod$\acute{y}$m Property for $1<p< \infty$, $L_p(\mu, Y)$ has property-$U$ (property-$SU$) in $L_p(\mu, X)$ if and only if $Y$ is so in $X$. We show that if $Z\subseteq Y\subseteq X$, where $Y$ has property-$U$ ($SU$) in $X$ then $Y/Z$ has property-$U$ ($SU$) in $X/Z$. On the other hand $Y$ has property-$SU$ in $X$ if $Y/Z$ has property-$SU$ in $X/Z$ and $Z (\subseteq Y)$ is an M-ideal in $X$. It is observed that a smooth Banach space of dimension $>3$ is a Hilbert space if and only if for any two subspaces $Y, Z$ with property-$SU$ in $X$, $Y+Z$ has property-$SU$ in $X$ whenever $Y+Z$ is closed. We characterize all hyperplanes in $c_0$ which have property-$SU$.