# A generalization of Gelfand-Naimark-Stone duality to completely regular spaces

## TL;DR

This paper extends the Gelfand-Naimark-Stone duality from compact Hausdorff spaces to completely regular spaces by introducing basic extensions of $ ext{l}$-algebras and establishing a duality with compactifications.

## Contribution

It generalizes the classical duality to a broader class of spaces through the development of basic extensions and maximal basic extensions.

## Key findings

- Established duality between maximal basic extensions and Stone-ech compactifications.
- Extended Gelfand-Naimark-Stone duality to completely regular spaces.
- Connected the category of completely regular spaces with a category of algebraic structures.

## Abstract

Gelfand-Naimark-Stone duality establishes a dual equivalence between the category ${\sf KHaus}$ of compact Hausdorff spaces and the category ${\boldsymbol{\mathit{uba}\ell}}$ of uniformly complete bounded archimedean $\ell$-algebras. We extend this duality to the category ${\sf CReg}$ of completely regular spaces. This we do by first introducing basic extensions of bounded archimedean $\ell$-algebras and generalizing Gelfand-Naimark-Stone duality to a dual equivalence between the category ${\boldsymbol{\mathit{ubasic}}}$ of uniformly complete basic extensions and the category ${\sf C}$ of compactifications of completely regular spaces. We then introduce maximal basic extensions and prove that the subcategory ${\boldsymbol{\mathit{mbasic}}}$ of ${\boldsymbol{\mathit{ubasic}}}$ consisting of maximal basic extensions is dually equivalent to the subcategory ${\sf SComp}$ of ${\sf Comp}$ consisting of Stone-\v{C}ech compactifications. This yields the desired dual equivalence for completely regular spaces since ${\sf CReg}$ is equivalent to ${\sf SComp}$.