Order-preserving unique Hahn-Banach extensions

[2504.03386] Order-preserving unique Hahn-Banach extensions Skip to main content Search Submit Donate Log in Mathematics > Functional Analysis arXiv:2504.03386 (math) [Submitted on 4 Apr 2025 ( v1 ), last revised 28 Aug 2026 (this version, v2)] Title: Order-preserving unique Hahn-Banach extensions Authors: Tanmoy Paul , T.S.S.R.K. Rao View PDF HTML (experimental) 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. Subjects: Functional Analysis (math.FA) MSC classes: 41A50, 46B20, 46E15, 46B25, 46G10, 46L05 Cite as: arXiv:2504.03386 [math.FA] (or arXiv:2504.03386v2 [math.FA] for this version) https://doi.org/10.48550/arXiv.2504.03386 Focus to learn more arXiv-issued DOI via DataCite Related DOI : https://doi.org/10.1007/s44146-026-00269-y Focus to learn more DOI(s) linking to related resources Submission history From: Tanmoy Paul [ view email ] [v1] Fri, 4 Apr 2025 11:56:38 UTC (13 KB) [v2] Fri, 28 Aug 2026 10:01:26 UTC (14 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.FA < prev | next > new | recent | 2025-04 Change to browse by: math References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from (opens in new tab) (opens in new tab) (opens in new tab)

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