Continuous mappings with null support

Abstract

Let $X$ be a (topological) space and let $\mathcal{I}$ be an ideal in $X$, that is, a collection of subsets of $X$ which contains all subsets of its elements and is closed under finite unions. The elements of $\mathcal{I}$ are called null. The space $X$ is locally null if each $x$ in $X$ has a null neighborhood in $X$. Let $C_b(X)$ denote the normed algebra of all continuous bounded real-valued mappings on $X$ equipped with the supremum norm, $C_0(X)$ denote the subalgebra of $C_b(X)$ consisting of elements vanishing at infinity and $C_{00}(X)$ the subalgebra of $C_b(X)$ consisting of elements with compact support. We study the normed subalgebra $C_{00}^{\mathcal{I}}(X)$ of $C_b(X)$ consisting of all $f$ in $C_b(X)$ whose support has a null neighborhood in $X$, and the Banach subalgebra $C_0^{\mathcal{I}}(X)$ of $C_b(X)$ consisting of all $f$ in $C_b(X)$ such that $|f{|}^{-1}([1/n,\infty ))$ has a null neighborhood in $X$ for all positive integer $n$. We prove that if $X$ is a normal locally null space then $C_{00}^{\mathcal{I}}(X)$ and $C_0^{\mathcal{I}}(X)$ are respectively isometrically isomorphic to $C_{00}(Y)$ and $C_0(Y)$ for a unique locally compact Hausdorff space $Y$; furthermore, $C_{00}^{\mathcal{I}}(X)$ is dense in $C_0^{\mathcal{I}}(X)$. We construct $Y$ explicitly as a subspace of the Stone--Čech compactification $\beta X$ of $X$. The space $Y$ is locally compact (and countably compact, in certain cases), contains $X$ densely, and in specific cases turns out to be familiar subspaces of $\beta X$. The known topological structure of $Y$ enables us to establish several commutative Gelfand--Naimark type theorems and derive results not generally expected to be deducible from the standard Gelfand theory.

53 pages

References in corpus (4)

• One-point extensions of locally compact paracompact spaces
• Connectedness modulo a topological property
• Representations of certain normed algebras
• Representations of certain Banach algebras

Cited by in corpus (2)

• Representations of certain normed algebras
• Representations of certain Banach algebras