2022/10/27 by Marco Abbadini, Vincenzo Marra, Abbadini, Marco +3 · 1 citation
Mathematics · #06F20 (Primary) #54A05 #54C30 (Secondary) #Advanced Banach Space Theory #Advanced Topology and Set Theory #FOS: Mathematics #Functional Analysis (math.FA) #Homotopy and Cohomology in Algebraic Topology #Logic (math.LO)
paper · pdf · doi:10.48550/arxiv.2210.15341
openalex publication_date 2022/10/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We extend Yosida's 1941 version of Stone-Gelfand duality to metrically complete unital lattice-ordered groups that are no longer required to be real vector spaces. This calls for a generalised notion of compact Hausdorff space whose points carry an arithmetic character to be preserved by continuous maps. The arithmetic character of a point is (the complete isomorphism invariant of) a metrically complete additive subgroup of the real numbers containing 1, namely, either (1)/(n)ℤ for an integer n = 1, 2, …, or the whole of ℝ. The main result needed to establish the extended duality theorem is a substantial generalisation of Urysohn's Lemma to such "arithmetic" compact Hausdorff spaces. The original duality is obtained by considering the full subcategory of spaces whose each point is assigned the entire group of real numbers. In the introduction we indicate motivations from and connections with the theory of dimension groups.