[論文レビュー] Generalizing Gelfand duality to Nachbin spaces

G. Bezhanishvili, P. J. Morandi|arXiv (Cornell University)|Jan 15, 2026

Advanced Operator Algebra Research被引用数 0

ひとこと要約

The paper extends Gelfand duality from compact Hausdorff spaces to Nachbin spaces by introducing Nachbin proximities on bounded archimedean ℓ-algebras, yielding dualities with categories of Nachbin spaces and uniformly complete sbal-/baℓ-algebras.

ABSTRACT

We introduce the notion of a Nachbin proximity on a bounded archimedean $\ell$-algebra (bal-algebra), and show that Gelfand duality lifts to yield a dual equivalence between the category of uniformly complete bal-algebras equipped with a closed Nachbin proximity and that of Nachbin spaces (compact ordered spaces). The key ingredients of the proof include appropriate generalizations of the Stone-Weierstrass theorem and Dieudonné's lemma. We also develop an alternate approach by means of bounded archimedean $\ell$-semialgebras (sbal-algebras), from which we derive De Rudder--Hansoul duality.

研究の動機と目的

• Motivate and formalize the inclusion of order in Gelfand duality via Nachbin spaces.
• Introduce Nachbin proximities on baℓ-algebras and related sbal-algebras to axiomatize continuous order-preserving functions.
• Establish dualities between Nachbin spaces and algebraic structures (unbaℓ and usbal) using generalized Stone-Weierstrass and Dieudonné-type results.

提案手法

• Define and study proximities on baℓ-algebras and reflexive proximities called Nachbin proximities.
• Construct functors between sbal, pbaℓ, and baℓ to relate sbal and spbaℓ via adjunctions.
• Introduce sbal-envelopes and prove baℓ is a reflective subcategory of sbal; obtain De Rudder–Hansoul-like dualities.
• Develop a generalized Stone-Weierstrass theorem for Nachbin spaces to achieve density results.
• Prove a Nachbin version of Dieudonné’s lemma to lift Gelfand duality to Nachbin spaces.
• Demonstrate contravariant adjunctions between Nach and nbaℓ/nb aℓ, leading to dualities.

実験結果

リサーチクエスチョン

• RQ1 How can Gelfand duality be extended to Nachbin spaces?
• RQ2 What algebraic structures (with proximities) correspond dually to Nachbin spaces?
• RQ3 Can Stone-Weierstrass and Dieudonné’s lemma be generalized to Nachbin spaces to lift dualities?
• RQ4 What are the equivalent categorical presentations (baℓ-, sbal-, sbal/skeletal) that realize Nachbin duality?

主な発見

• There is a dual equivalence between the category of uniformly complete baℓ-algebras equipped with a closed Nachbin proximity and the category of Nachbin spaces.
• A skeletal subcategory spbaℓ of proximity baℓ-algebras corresponds to sbal via a left adjoint–right adjoint equivalence.
• The C≤(X) construction embeds as the fixpoints of the Nachbin proximity on C(X), linking order-preserving functions to Proximity-based structure.
• Two generalized Gelfand dualities are obtained: one via Nachbin proximities on baℓ-algebras and another via sbal-algebras, with the latter yielding De Rudder–Hansoul duality.
• A uniformly dense image result (Nachbin Stone-Weierstrass) and a Dieudonné-type lemma are established in the Nachbin setting, enabling dualities with Nachbin spaces.
• The framework recovers and extends the De Rudder–Hansoul approach while also aligning with existing work on proximity-based Gelfand dualities.