arxiv: 2605.03709 · v1 · submitted 2026-05-05 · 🧮 math.FA

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Kadison duality for partially convex sets

Tea \v{S}trekelj

Pith reviewed 2026-05-07 13:18 UTC · model grok-4.3

classification 🧮 math.FA

keywords Kadison dualitypartially convex setsfree order unit modulescategorical dualityArchimedean order unitpartially affine functionsC*-algebras

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The pith

Compact regular partially convex sets are categorically dual to free order unit modules

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper extends the Kadison duality between compact convex sets and function systems to the setting of partial convexity. A partially convex set is convex only when certain designated variables are varied while others are held fixed. The work introduces regular partially convex sets and proves they stand in categorical duality with free order unit modules, which are finitely generated free modules over commutative C*-algebras equipped with a compatible Archimedean order unit structure. If the duality holds, questions about geometry in the partial setting can be translated into algebraic questions about these modules, and theorems about the modules can be read back as statements about the sets. The paper also establishes that partially affine polynomials are dense in the continuous partially affine functions and that compact partially convex sets can be separated from outer points by a Hahn-Banach-type theorem.

Core claim

The central claim is a categorical duality between compact regular partially convex sets and free order unit modules. For any compact regular partially convex set K, the space of continuous functions on K that are affine in the convex variables forms a free order unit module. Conversely, the partially convex state space of any free order unit module is a compact regular partially convex set. The duality is an equivalence of categories that sends morphisms to morphisms in the opposite direction.

What carries the argument

The notion of a regular partially convex set, whose continuous partially affine functions form a free order unit module (a finitely generated free module over a commutative C*-algebra with compatible Archimedean order unit structure) that encodes the partial convexity.

If this is right

The space of continuous functions on a compact regular partially convex set that are affine in the convex variables is a free order unit module.
Partially affine polynomials are dense in the space of continuous partially affine functions on any compact regular partially convex set.
Compact partially convex sets can be separated from their outer points by a Hahn-Banach-type theorem.
The correspondence is an equivalence of categories.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The density theorem supplies a concrete approximation tool that can be used to reduce problems about continuous partially affine functions to algebraic questions in the dual module.
The separation result may be applied directly to obtain new existence statements for supporting functionals in settings where only some variables are convex.
Because the duality is categorical, any functorial construction on free order unit modules automatically yields a corresponding construction on compact regular partially convex sets.

Load-bearing premise

The assumption that a partially convex set is regular is load-bearing, because it is what forces the dual space of continuous partially affine functions to be a finitely generated free module with the required Archimedean order unit structure.

What would settle it

A compact partially convex set that satisfies the regularity condition yet whose space of continuous partially affine functions fails to be a free module over a commutative C*-algebra with Archimedean order unit would show the claimed duality does not hold.

Figures

Figures reproduced from arXiv: 2605.03709 by Tea \v{S}trekelj.

**Figure 1.** Figure 1: A partially convex set, bounded by the lines y = ±1 and the hyperbola 2x 2 = y 2 + 1. 1.2. Main results. The main contribution of this paper is the extension of Kadison’s duality between convex sets and function systems to the setting of partial convexity. Precisely, we focus on regular partially convex sets as introduced in Definition 3.2, where regularity ensures the wellbehaved nature of the set’s boun… view at source ↗

read the original abstract

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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This extends Kadison duality to partial convexity with an explicit categorical equivalence, but regularity may limit its reach.

read the letter

The main thing to know about this paper is that it sets up a categorical duality between compact regular partially convex sets and free order unit modules, extending Kadison's classical result on convex sets and function systems to the partial convexity case. They introduce the notion of regularity for these sets and show that the continuous partially affine functions give a finitely generated free module over a commutative C*-algebra with Archimedean order unit. Going the other way, the partially convex states recover the original set. The duality is backed by a density theorem showing partially affine polynomials are dense in the continuous ones, and a separation result that separates compact partially convex sets from points outside them. The stress test confirms the functors work without inconsistencies or extra hidden assumptions. This is done well because the constructions are explicit and follow from standard C* and order theory without forcing the partial aspect into a full convexity mold. No circularity in the definitions. The soft spot is mainly around how broad the regular partially convex sets are. Regularity is the key assumption that makes everything work, so if it is overly strong it could limit the applicability. It would help to have more examples showing what kinds of sets satisfy it and what this buys in applications like optimization. This kind of paper is for people in functional analysis who follow duality results and categorical reformulations in convex geometry. It should interest those looking at mixed convexity settings. The work shows clear thinking and honest use of the literature, so it deserves to go to peer review for a full check of the proofs.

Referee Report

0 major / 3 minor

Summary. The paper extends Kadison duality from compact convex sets and function systems to the setting of partial convexity. It introduces the notions of regular partially convex sets and free order unit modules (finitely generated free modules over commutative C*-algebras equipped with compatible Archimedean order-unit structures). Explicit functors are constructed in both directions: the space of continuous partially affine functions on a compact regular partially convex set K yields the dual module, while the partially convex state space of a free order unit module recovers a compact regular partially convex set. The main result is a categorical equivalence between these two categories. Supporting results include a Stone-Weierstrass-type density theorem showing that partially affine polynomials are dense in the continuous partially affine functions, and a Hahn-Banach-type separation theorem for compact partially convex sets.

Significance. If the constructions and proofs are correct, the result provides a clean categorical duality that generalizes a classical theorem in functional analysis to a broader class of partially convex structures. The explicit functors, the density theorem, and the separation result are concrete tools that could be applied in contexts where convexity holds only in selected variables, such as certain optimization problems or noncommutative settings. The paper ships explicit functorial constructions and supporting lemmas that close the equivalence, which strengthens the contribution.

minor comments (3)

The definition of a 'regular' partially convex set (introduced early in the paper) is load-bearing for the duality; a short paragraph clarifying why this regularity condition is preserved under the state-space functor and is strictly weaker than full convexity would improve readability.
In the statement of the Stone-Weierstrass-type theorem, the precise algebra of 'partially affine polynomials' should be defined before the density claim, including how the partial convexity variables interact with the polynomial ring.
The Hahn-Banach-type separation theorem is stated for 'outer points'; a brief remark on whether the separation is strict or weak, and how it reduces to the classical case when all variables are convex, would help readers connect to prior work.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary and significance assessment of our manuscript extending Kadison duality to partially convex sets. The recommendation for minor revision is noted. No specific major comments were provided in the report, so we have no points requiring point-by-point rebuttal or revision.

Circularity Check

0 steps flagged

No circularity detected in derivation chain

full rationale

The paper defines new objects (regular partially convex sets and free order unit modules) and constructs explicit functors establishing a categorical duality, supported by standard results such as Stone-Weierstrass density and Hahn-Banach separation. These steps rely on external functional analysis theorems rather than self-referential definitions, fitted parameters renamed as predictions, or load-bearing self-citations. The central equivalence is built from independent constructions in both directions without reducing to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 2 invented entities

The duality rests on the newly introduced definitions of regularity for partially convex sets and the free order unit module structure; these are not drawn from prior literature but defined within the paper to enable the correspondence.

axioms (2)

standard math Standard properties of commutative C*-algebras and Archimedean order unit spaces
The dual objects are built directly on these established structures from operator algebra theory.
domain assumption Existence and compatibility of the partial convexity and regularity conditions
The paper introduces regularity as the key assumption that makes the duality and the state space correspondence hold.

invented entities (2)

regular partially convex set no independent evidence
purpose: Objects for which the extended duality is stated
New definition extending standard convex sets to allow convexity only in designated variables.
free order unit module no independent evidence
purpose: Algebraic dual to the regular partially convex sets
New module structure over C*-algebras with order unit, defined to match the geometric side.

pith-pipeline@v0.9.0 · 5491 in / 1583 out tokens · 66798 ms · 2026-05-07T13:18:45.987829+00:00 · methodology

discussion (0)

Reference graph

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