The Banach algebra of continuous bounded functions with separable support

TL;DR

This paper establishes that the algebra of bounded continuous functions with separable support on a locally separable metrizable space is a Banach algebra isometrically isomorphic to a space of continuous functions vanishing at infinity on a uniquely constructed locally compact Hausdorff space, extending classical Gelfand theory.

Contribution

It proves a Gelfand--Naimark type theorem for $C_s(X)$, explicitly constructs the spectrum space $Y$, and determines the algebra's dimension, providing new insights into the structure of these function spaces.

Findings

$C_s(X)$ is a Banach algebra isometrically isomorphic to $C_0(Y)$.

The spectrum space $Y$ is explicitly constructed as a subspace of the Stone--ech compactification.

The algebra $C_s(X)$ has finite or infinite dimension depending on $X$, and $Y$ is countably compact, non-normal if $X$ is non-separable.

Abstract

We prove a commutative Gelfand--Naimark type theorem, by showing that the set $C_s(X)$ of continuous bounded (real or complex valued) functions with separable support on a locally separable metrizable space $X$ (provided with the supremum norm) is a Banach algebra, isometrically isomorphic to $C_0(Y)$, for some unique (up to homeomorphism) locally compact Hausdorff space $Y$. The space $Y$, which we explicitly construct as a subspace of the Stone--\v{C}ech compactification of $X$, is countably compact, and if $X$ is non-separable, is moreover non-normal; in addition $C_0(Y)=C_{00}(Y)$. When the underlying field of scalars is the complex numbers, the space $Y$ coincides with the spectrum of the ${C}^*$-algebra $C_s(X)$. Further, we find the dimension of the algebra $C_s(X)$.