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

Gelfand-Naimark-Stone duality establishes a dual equivalence between the category $\mathsf{K}\mathsf{H}\mathsf{a}\mathsf{u}\mathsf{s}$ of compact Hausdorff spaces and the category $$ of uniformly complete bounded archimedean $\mathrm{\ell }$-algebras. We extend this duality to the category $\mathsf{C}\mathsf{R}\mathsf{e}\mathsf{g}$ of completely regular spaces. This we do by first introducing basic extensions of bounded archimedean $\mathrm{\ell }$-algebras and generalizing Gelfand-Naimark-Stone duality to a dual equivalence between the category $$ 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 $$ of $$ 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 ${\sf CReg}$isequivalentto${\sf SComp}$.