Ideals in CB(X) arising from ideals in X

TL;DR

This paper explores the relationship between set-theoretic ideals of a topological space and algebraic ideals in the algebra of bounded continuous functions, providing representation theorems via Stone--Čech compactification.

Contribution

It introduces a novel topological approach to represent and analyze ideals in $C_B(X)$ using subspaces of the Stone--Čech compactification, linking algebraic and topological structures.

Findings

Representation theorems for ideals of $C_B(X)$

Characterization of spectra of closed ideals

Examples illustrating the structure of ideals

Abstract

Let $X$ be a completely regular topological space. We assign to each (set theoretic) ideal of $X$ an (algebraic) ideal of $C_B(X)$, 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 $C_B(X)$. This is done by associating a certain subspace of the Stone--\v{C}ech compactification $\beta X$ of $X$ to each ideal of $X$. This subspace of $\beta X$ has a simple representation, and in the case when the assigned ideal of $C_B(X)$ is closed, coincides with its spectrum as a $C^*$-subalgebra of $C_B(X)$. This in particular provides information about the spectrum of those closed ideals of $C_B(X)$ which have such representations. This includes the non-vanishing closed ideals of $C_B(X)$ whose spectrums are studied in great detail. Our representation theorems help to understand the structure of certain ideals of $C_B(X)$. 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--\v{C}ech compactification.