Uniqueness of Hahn--Banach extensions and some of its variants

TL;DR

This paper explores the conditions under which Hahn--Banach extensions are unique or have variants, providing new characterizations and examining classical Banach spaces and tensor product contexts.

Contribution

It introduces new characterizations of properties related to Hahn--Banach extension uniqueness and analyzes their occurrence in classical Banach spaces and tensor product spaces.

Findings

A hyperplane in c_0 has property-(HB) iff it is an M-summand.

An isometry in L(X,Y*) has a unique norm-preserving extension if Y has property-(SU).

Finite-dimensional subspaces of c_0 have property-(k-U), and their duals are k-strictly convex in certain cases.

Abstract

In this study, we analyze the various strengthening and weakening of the uniqueness of the Hahn--Banach extension. In addition, we consider the case in which $Y$ is an ideal of $X$. In this context, we study the property-$(U)/ (SU)/ (HB)$ and property-$(k-U)$ for a subspace $Y$ of a Banach space $X$. We obtain various new characterizations of these properties. We discuss various examples in the classical Banach spaces, where the aforementioned properties are satisfied and where they fail. It is observed that a hyperplane in $c_0$ has property-$(HB)$ if and only if it is an $M$-summand. Considering $X, Z$ as Banach spaces and $Y$ as a subspace of $Z$, by identifying $(X\widehat{\otimes}_\pi Y)^*\cong \mathcal{L}(X,Y^*)$, we observe that an isometry in $\mathcal{L}(X,Y^*)$ has a unique norm-preserving extension over $(X\widehat{\otimes}_\pi Z)$ if $Y$ has property-$(SU)$ in $Z$. It is observed that a finite dimensional subspace $Y$ of $c_0$ has property-$(k-U)$ in $c_0$, and if $Y$ is an ideal, then $Y^*$ is a $k$-strictly convex subspace of $\ell_1$ for some natural $k$.