On closed non-vanishing ideals in CB(X)

TL;DR

This paper investigates the structure of non-vanishing closed ideals in the algebra of bounded continuous functions on a topological space, linking algebraic properties to topological characteristics of an associated space.

Contribution

It provides necessary and sufficient algebraic conditions for these ideals to correspond to spaces with various connectedness properties.

Findings

Characterizes when the associated space is locally connected.

Identifies conditions for total disconnectedness and zero-dimensionality.

Links algebraic properties of ideals to topological features of the associated space.

Abstract

Let $X$ be a completely regular topological space. We study closed ideals $H$ of $C_B(X)$, the normed algebra of bounded continuous scalar-valued mappings on $X$ equipped with pointwise addition and multiplication and the supremum norm, which are non-vanishing, in the sense that, there is no point of $X$ at which every element of $H$ vanishes. This is done by studying the (unique) locally compact Hausdorff space $Y$ associated to $H$ in such a way that $H$ and $C_0(Y)$ are isometrically isomorphic. We are interested in various connectedness properties of $Y$. In particular, we present necessary and sufficient (algebraic) conditions for $H$ such that $Y$ satisfies (topological) properties such as locally connectedness, total disconnectedness, zero-dimensionality, strong zero-dimensionality, total separatedness or extremal disconnectedness.