Kadison duality for partially convex sets

@misc{strekelj2026kadison,
  title = {{Kadison duality for partially convex sets}},
  author = {Tea Štrekelj},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2605.03709},
  eprint = {2605.03709},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.03709},
}

TY  - UNPB
TI  - Kadison duality for partially convex sets
AU  - Tea Štrekelj
PY  - 2026
DA  - 2026/09/11
PB  - arXiv
DO  - 10.48550/arxiv.2605.03709
UR  - https://arxiv.org/abs/2605.03709
AB  - 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 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.
ER  -