Order-preserving unique Hahn-Banach extensions

Abstract

Let $X$ be a real Banach lattice with a unit, and let $Y \subseteq X$ be a closed subspace containing the unit. In this paper, we study the order-theoretic (also isometric) structure of $Y$ that it may inherit from $X$ under some additional conditions. Specifically, we consider the case where every continuous positive linear functional in the unit sphere of $Y^\ast$ admits a unique positive norm-preserving extension in $X^\ast$. Our results depend on the specific nature of the embedding of $Y$ in $X$. For a compact convex set $K$ with closed extreme boundary $\partial_e K$, for the restriction isometry of $A(K)$ (which is also order-preserving) into $C(\partial_e K)$, uniqueness of extensions of positive functionals leads to $K$ being a simplex and the restriction embedding being onto. In contrast, for a Choquet simplex $K$, we show that under the canonical embedding in the bidual $A(K)^{\ast\ast}$ (which is an abstract $M$-space), positive linear functionals having unique positive linear extensions imply that $K$ is a finite-dimensional simplex.