Bounded Archimedean ℓ-algebras and Gelfand-Neumark-Stone duality

Abstract

By Gelfand-Neumark duality, the category C* Alg of commutative C* algebras is dually equivalent to the category of compact Hausdorff spaces, which by Stone duality, is also dually equivalent to the category ubal of uniformly complete bounded Archimedean ℓ-algebras. Consequently, C*Alg is equivalent to ubal, and this equivalence can be described through complexification. In this article we study ubal within the larger category bal of bounded Archimedean ℓ-algebras. We show that ubal is the smallest nontrivial reflective subcategory of bal, and that ubal consists of exactly those objects in bal that are epicomplete, a fact that includes a categorical formulation of the Stone-Weierstrass theorem for bal. It follows that ubal is the unique nontrivial reflective epicomplete subcategory of bal. We also show that each nontrivial reflective subcategory of bal is both monoreflective and epireflective, and exhibit two other interesting reflective subcategories of bal involving Gelfand rings and square closed rings. Dually, we show that Specker R-algebras are precisely the co-epicomplete objects in bal. We prove that the category spec of Specker R-algebras is a mono-coreflective subcategory of bal that is co-epireflective in a mono-coreflective subcategory of bal consisting of what we term ℓ-clean rings, a version of clean rings adapted to the ordertheoretic setting of bal. We conclude the article by discussing the import of our results in the setting of complex *-algebras through complexification. © Guram Bezhanishvili, Patrick J. Morandi, Bruce Olberding, 2013.