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

Abstract

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

26 pages; comments welcome