Gelfand-Naimark-Stone duality for normal spaces and insertion theorems

TL;DR

This paper extends the Gelfand-Naimark-Stone duality to characterize normal, Lindelöf, and locally compact Hausdorff spaces using algebraic methods, offering new insights into classical theorems like Katětov-Tong and Stone-Weierstrass.

Contribution

It generalizes the duality framework to broader classes of spaces and provides an algebraic perspective on classical topological theorems.

Findings

Characterization of normal, Lindelöf, and locally compact spaces via algebraic duality.

New algebraic proofs of classical theorems such as Katětov-Tong and Stone-Weierstrass.

Extension of duality to non-compact spaces using uniformly complete bounded archimedean ℓ-algebras.

Abstract

Gelfand-Naimark-Stone duality provides an algebraic counterpart of compact Hausdorff spaces in the form of uniformly complete bounded archimedean $\ell$-algebras. In [4] we extended this duality to completely regular spaces. In this article we use this extension to characterize normal, Lind\"{e}lof, and locally compact Hausdorff spaces. Our approach gives a different perspective on the classical theorems of Kat\v{e}tov-Tong and Stone-Weierstrass.