2018/12/18 by Bezhanishvili, Guram, Morandi, Patrick J., Olberding, Bruce
#06F25 #13J25 #54C30 #54D15 #54D35 #Category Theory (math.CT) #FOS: Mathematics #General Topology (math.GN)
paper · doi:10.48550/arxiv.1812.07599
Gelfand-Naimark-Stone duality establishes a dual equivalence between the category \sf KHaus of compact Hausdorff spaces and the category \boldsymbolubaℓ of uniformly complete bounded archimedean ℓ-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 ℓ-algebras and generalizing Gelfand-Naimark-Stone duality to a dual equivalence between the category \boldsymbolubasic 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 \boldsymbolmbasic of \boldsymbolubasic consisting of maximal basic extensions is dually equivalent to the subcategory \sf SComp of \sf Comp consisting of Stone-Čech compactifications. This yields the desired dual equivalence for completely regular spaces since \sf CReg is equivalent to \sf SComp.