Order-preserving unique Hahn-Banach extensions

TL;DR

The paper addresses how order-theoretic and isometric structures are inherited from a Banach space $X$ to a closed subspace $Y$ under the unique extension property for positive functionals. It distinguishes two embedding contexts for $$A(K)$$ spaces: (i) restriction embeddings into $C(ully ext{decorated})$ and (ii) canonical bidual embeddings for Choquet simplexes, deriving sharp geometric consequences on the underlying compact convex set $K$—namely Bauer simplex status and onto-ness in the restriction case, and finite dimensionality in the bidual Choquet case. The results connect order-theoretic Hahn–Banach extension properties with concrete geometric/classification outcomes for $K$ (Bauer vs. finite-dimensional Choquet/simplex) and provide operator-algebraic analogues, including a structural characterization of weakly Hahn–Banach smooth C*-algebras as $c_0$-sums of compact operators. Overall, the work offers an order-theoretic analogue to known C*-algebraic results, clarifying how extension properties shape the geometry of the dual and bidual spaces and the associated simplex structure.

Abstract

Let $X$ be a real Banach lattice with a unit, 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. One such condition is to assume that all continuous positive linear functionals in the unit sphere of $Y^\ast$ have unique positive norm preserving extensions in $X^\ast$. Our answers 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. On the other hand, for a Choquet simplex $K$, under the canonical embedding in the bidual $A(K)^{\ast\ast}$ (which is an abstract $M$-space) uniqueness of extensions implies that $K$ is a finite dimensional simplex. This gives an order theoretic analogue of a result of Contreras, Pay$\acute{a}$ and Werner, proved in the context of unital $C^\ast$-algebras.

Theorems & Definitions (25)

• Definition 1.1
• Definition 2.1
• Theorem 3.1
• proof
• Lemma 3.2
• proof
• Example 3.3
• Theorem 3.4
• proof
• Corollary 3.5