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

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 $(X{\hat{\otimes }}_{\pi }Y{)}^{\ast }\cong \mathcal{L}(X,Y^{\ast })$, we observe that an isometry in $\mathcal{L}(X,Y^{\ast })$ has a unique norm-preserving extension over $(X{\hat{\otimes }}_{\pi }Z)$ 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 $Y^{\ast }$ is a $k$-strictly convex subspace of ${\ell }_1$ for some natural $k$.