Phelps Property U and $C(K)$ spaces

TL;DR

This paper systematically studies Property U in Banach spaces, focusing on U-embeddings into $C(K)$ spaces, and explores conditions for their existence in various settings.

Contribution

It introduces the concept of U-embeddings and develops a framework for their analysis, especially into $C(K)$ spaces, extending prior understanding of Property U.

Findings

Established criteria for U-embeddings into $C(K)$ spaces.

Analyzed U-embeddings for finite-dimensional spaces.

Provided results for general Banach spaces.

Abstract

A subspace $X$ of a Banach space $Y$ has $\textit{Property U}$ whenever every continuous linear functional on $X$ has a unique norm-preserving (i.e., Hahn$-$Banach) extension to $Y$ (Phelps, 1960). Throughout this document we introduce and develop a systematic study of the existence of $\textit{U-embeddings}$ between Banach spaces $X$ and $Y$, that is, isometric embeddings of $X$ into $Y$ whose ranges have property U. In particular, we are interested in the case that $Y=C(K)$, where $K$ is a compact Hausdorff topological space. We provide results for general Banach spaces and for some specific set-ups, such as $X$ being a finite-dimensional space or a $C(K)$-space.