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

Guram Bezhanishvili ; Patrick J. Morandi ; Bruce Olberding

Theory and applications of categories, Tome 28 (2013), pp. 435-475

Résumé

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 $\ell$-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 $\ell$-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 ${\mathbb R}$-algebras are precisely the co-epicomplete objects in $bal$. We prove that the category $spec$ of Specker $\mathbb R$-algebras is a mono-coreflective subcategory of $bal$ that is co-epireflective in a mono-coreflective subcategory of $bal$ consisting of what we term $\ell$-clean rings, a version of clean rings adapted to the order-theoretic setting of $bal$.

We conclude the article by discussing the import of our results in the setting of complex $*$-algebras through complexification.

Détail

Publié le : 2013-07-15

Classification : 06F25, 13J25, 54C30
Keywords: Ring of continuous real-valued functions, l-ring, l-algebra, uniform completeness, Stone-Weierstrass theorem, commutative $C^*$-algebra, compact Hausdorff space, Gelfand-Neumark-Stone duality