2024/06/18 by Matthias Schötz, Schötz, Matthias
Mathematics · #06F25 #13J25 #16G30 #47L60 #Advanced Topics in Algebra #FOS: Mathematics #Operator Algebras (math.OA) #Rings and Algebras (math.RA) #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2406.13063
openalex publication_date 2024/06/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The following representation theorem is proven: A partially ordered commutative ring R is a subring of a ring of almost everywhere defined continuous real-valued functions on a compact Hausdorff space X if and only if R is archimedean and localizable. Here we assume that the positive cone of R is closed under multiplication and stable under multiplication with squares, but actually one of these assumptions implies the other. An almost everywhere defined function on X is one that is defined on a dense open subset of X. A partially ordered commutative ring R is archimedean if the underlying additive partially ordered abelian group is archimedean, and R is localizable essentially if its order is compatible with the construction of a localization with sufficiently large, positive denominators. As applications we discuss the σ-bounded case, lattice-ordered commutative rings (f-rings), partially ordered fields, and commutative operator algebras.