Representations of certain Banach algebras

TL;DR

This paper characterizes certain Banach subalgebras of bounded continuous functions as isomorphic to function spaces on specific locally compact spaces, using explicit constructions related to the Stone–Čech compactification.

Contribution

It provides an explicit construction of the space Y for which subalgebras of C_b(X) are isometrically isomorphic to C_0(Y), extending standard Gelfand theory.

Findings

Explicit construction of Y as a subspace of Stone–Čech compactification

Identification of isometric isomorphisms between subalgebras and C_0(Y)

New insights into properties of H and Y beyond Gelfand theory

Abstract

For a space $X$ denote by $C_b(X)$ the Banach algebra of all continuous bounded scalar-valued functions on $X$ and denote by $C_0(X)$ the set of all elements in $C_b(X)$ which vanish at infinity. We prove that certain Banach subalgebras $H$ of $C_b(X)$ are isometrically isomorphic to $C_0(Y)$, for some unique (up to homeomorphism) locally compact Hausdorff space $Y$. The space $Y$ is explicitly constructed as a subspace of the Stone--\v{C}ech compactification of $X$. The known construction of $Y$ enables us to examine certain properties of either $H$ or $Y$ and derive results not expected to be deducible from the standard Gelfand Theory.