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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2605.03709 |
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| _version_ | 1866915980961120256 |
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| author | Štrekelj, Tea |
| author_facet | Štrekelj, Tea |
| contents | This paper extends the Kadison duality between compact convex sets and function systems to the setting of partial convexity. A partially convex set is a set that is convex in a designated set of convex variables when the others are held fixed. We introduce the notion of a regular partially convex set and identify its dual as a finitely generated free module over a commutative C*-algebra endowed with a compatible Archimedean order unit structure. We call such spaces free order unit modules. We prove that for any compact regular partially convex set K, the space of continuous functions on K that are affine in the convex variables is the canonical example of such a module. Conversely, we show that the partially convex state space of a free order unit module is a compact regular partially convex set. Our main result establishes a categorical duality between compact regular partially convex sets and free order unit modules. We also establish a Stone-Weierstrass-type theorem, demonstrating that partially affine polynomials are dense in the space of continuous partially affine functions on any compact regular partially convex set. Finally, we prove a Hahn-Banach-type separation theorem of compact partially convex sets from their outer points. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_03709 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Kadison duality for partially convex sets Štrekelj, Tea Functional Analysis This paper extends the Kadison duality between compact convex sets and function systems to the setting of partial convexity. A partially convex set is a set that is convex in a designated set of convex variables when the others are held fixed. We introduce the notion of a regular partially convex set and identify its dual as a finitely generated free module over a commutative C*-algebra endowed with a compatible Archimedean order unit structure. We call such spaces free order unit modules. We prove that for any compact regular partially convex set K, the space of continuous functions on K that are affine in the convex variables is the canonical example of such a module. Conversely, we show that the partially convex state space of a free order unit module is a compact regular partially convex set. Our main result establishes a categorical duality between compact regular partially convex sets and free order unit modules. We also establish a Stone-Weierstrass-type theorem, demonstrating that partially affine polynomials are dense in the space of continuous partially affine functions on any compact regular partially convex set. Finally, we prove a Hahn-Banach-type separation theorem of compact partially convex sets from their outer points. |
| title | Kadison duality for partially convex sets |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2605.03709 |