Bounded Archimedean `-algebras and Gelfand-neumark-stone Duality

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 uba` of uniformly complete bounded Archimedean `-algebras. Consequently, C∗Alg is equivalent to uba`, and this equivalence can be described through complexification. In this article we study uba` within the larger category ba` of bounded Archimedean `-algebras. We show that uba` is the smallest nontrivial reflective subcategory of ba`, and that uba` consists of exactly those objects in ba` that are epicomplete, a fact that includes a categorical formulation of the Stone-Weierstrass theorem for ba`. It follows that uba` is the unique nontrivial reflective epicomplete subcategory of ba`. We also show that each nontrivial reflective subcategory of ba` is both monoreflective and epireflective, and exhibit two other interesting reflective subcategories of ba` involving Gelfand rings and square closed rings. Dually, we show that Specker R-algebras are precisely the co-epicomplete objects in ba`. We prove that the category spec of Specker R-algebras is a mono-coreflective subcategory of ba` that is co-epireflective in a mono-coreflective subcategory of ba` consisting of what we term `-clean rings, a version of clean rings adapted to the ordertheoretic setting of ba`. We conclude the article by discussing the import of our results in the setting of complex ∗-algebras through complexification.