On closed non-vanishing ideals in C_B(X) I; Connectedness properties

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.

Introduction

Throughout this article by a space we mean a topological space. Completely regular spaces are assumed to be Hausdorff. The field of scalars, denoted by $\mathbb{F}$, is either $\mathbb{R}$ or $\mathbb{C}$ and is fixed throughout our discussion. For a space X we denote by $C_B(X)$ the set of all bounded continuous scalar-valued mappings on X. The set $C_B(X)$ is a Banach algebra when equipped with pointwise addition and multiplication and the supremum norm. We denote by $C_0(X)$ the normed subalgebra of $C_B(X)$ consisting of mappings which vanish at infinity.

The Banach algebra $C_B(X)$ plays a fundamental role in both topology and analysis. Here we assume the minimal assumption of complete regularity on the space X, and concentrate on the study of closed ideals H of $C_B(X)$ 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 which is associated to H in such a way that H and $C_0(Y)$ are isometrically isomorphic. The space Y is constructed by the second author in [8] as a subspace of the Stone–Čech compactification of X, and coincides with the spectrum of H (considered as a Banach algebra) in the case when the field of scalars is $\mathbb{C}$. The known (and simple) structure of Y enables us to study its properties. We are particularly interested in connectedness properties of Y. More specifically, we find 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.

Throughout this article we will make critical use of the theory of the Stone–Čech compactification. We state some basic properties of the Stone–Čech compactification in the following and refer the reader to the standard texts [4] and [11] for further reading on the subject.

The Stone–Čech compactification. Let X be completely regular space. By a compactification of X we mean a compact Hausdorff space αX which contains X as a dense subspace. Among compactifications of X there is the “largest” one called the Stone–Čech compactification (and denoted by βX) which is characterized by the property that every bounded continuous mapping $f:X\to \mathbb{F}$ is extendible to a continuous mapping $F:\beta X\to \mathbb{F}$. For a bounded continuous mapping $f:X\to \mathbb{F}$ we denote by $f_{\beta }$ or $f^{\beta }$ the (unique) continuous extension of f to βX.

The following are a few of the basic properties of βX. We use these properties throughout without explicitly referring to them.

• $\beta M=\beta X$ if $X\subseteq M\subseteq \beta X$.

• X is locally compact if and only if X is open in βX.

• ${\mathrm{cl}}_{\beta X}M$ is open-and-closed in βX if M is open-and-closed in X.