Order-preserving unique Hahn-Banach extensions
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910903570530304 |
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| author | Paul, Tanmoy Rao, T. S. S. R. K. |
| author_facet | Paul, Tanmoy Rao, T. S. S. R. K. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_03386 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Order-preserving unique Hahn-Banach extensions Paul, Tanmoy Rao, T. S. S. R. K. Functional Analysis 41A50, 46B20, 46E15, 46B25, 46G10, 46L05 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. |
| title | Order-preserving unique Hahn-Banach extensions |
| topic | Functional Analysis 41A50, 46B20, 46E15, 46B25, 46G10, 46L05 |
| url | https://arxiv.org/abs/2504.03386 |