Ideals in CB(X) arising from ideals in X

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--Čech 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 $C^{\ast }$-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--Čech compactification.

61 pages. This article contains (and extends) most theorems in arXiv:1302.2235

References in corpus (2)

• One-point extensions of locally compact paracompact spaces
• Connectedness modulo a topological property

Cited by in corpus (1)

• A Gelfand-Naimark type theorem