2015/08/31 by M. R. Koushesh, Koushesh, M. R.
Mathematics · #Advanced Topics in Algebra #Advanced Topology and Set Theory #FOS: Mathematics #Functional Analysis (math.FA) #Rings, Modules, and Algebras #math.FA
paper · pdf · doi:10.48550/arxiv.1508.07734
61 pages. This article contains (and extends) most theorems in arXiv:1302.2235
openalex publication_date 2015/08/31 · arxiv created 2016/06/07 · arxiv updated 2016/06/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let X be a completely regular topological space. We assign to each (set theoretic) ideal of X an (algebraic) ideal of CB(X), the normed algebra of continuous bounded complex valued mappings on X equipped with the supremum norm. We then prove several representation theorems for the assigned ideals of CB(X). This is done by associating a certain subspace of the Stone--Čech compactification βX of X to each ideal of X. This subspace of βX has a simple representation, and in the case when the assigned ideal of CB(X) is closed, coincides with its spectrum as a C^*-subalgebra of CB(X). This in particular provides information about the spectrum of those closed ideals of CB(X) which have such representations. This includes the non-vanishing closed ideals of CB(X) whose spectrums are studied in great detail. Our representation theorems help to understand the structure of certain ideals of CB(X). This has been illustrated by means of various examples. Our approach throughout will be quite topological and makes use of the theory of the Stone--Čech compactification.