2013/02/09 by M. R. Koushesh, Koushesh, M. R. · 1 citation
Mathematics · #16S60 #46E15 #46E25 #46H05 #46J10 #46J25 #54C35 #54D35 #54D60 #54D65 #Advanced Topics in Algebra #Advanced Topology and Set Theory #FOS: Mathematics #Functional Analysis (math.FA) #General Topology (math.GN) #Rings, Modules, and Algebras #math.FA #math.GN #msc:16S60 #msc:46E15 #msc:46E25 #msc:46H05 #msc:46J10 #msc:46J25 #msc:54C35 #msc:54D35 #msc:54D60 #msc:54D65
paper · pdf · doi:10.48550/arxiv.1302.2235
53 pages
openalex publication_date 2013/02/09 · arxiv created 2014/01/23 · arxiv updated 2014/01/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let X be a (topological) space and let \mathscr I be an ideal in X, that is, a collection of subsets of X which contains all subsets of its elements and is closed under finite unions. The elements of \mathscr I are called null. The space X is locally null if each x in X has a null neighborhood in X. Let Cb(X) denote the normed algebra of all continuous bounded real-valued mappings on X equipped with the supremum norm, C0(X) denote the subalgebra of Cb(X) consisting of elements vanishing at infinity and C00(X) the subalgebra of Cb(X) consisting of elements with compact support. We study the normed subalgebra C\mathscr I00(X) of Cb(X) consisting of all f in Cb(X) whose support has a null neighborhood in X, and the Banach subalgebra C\mathscr I0(X) of Cb(X) consisting of all f in Cb(X) such that |f|-1([1/n,∞)) has a null neighborhood in X for all positive integer n. We prove that if X is a normal locally null space then C\mathscr I00(X) and C\mathscr I0(X) are respectively isometrically isomorphic to C00(Y) and C0(Y) for a unique locally compact Hausdorff space Y; furthermore, C\mathscr I00(X) is dense in C\mathscr I0(X). We construct Y explicitly as a subspace of the Stone--Čech compactification βX of X. The space Y is locally compact (and countably compact, in certain cases), contains X densely, and in specific cases turns out to be familiar subspaces of βX. The known topological structure of Y enables us to establish several commutative Gelfand--Naimark type theorems and derive results not generally expected to be deducible from the standard Gelfand theory.