Spectra of Abelian C*-Subalgebra Sums

TL;DR

This paper investigates the spectral properties of sums of certain subalgebras of bounded continuous functions on locally compact spaces, revealing a twisted-sum topology structure and applications to compactifications.

Contribution

It characterizes the spectrum of sums of ideals and subalgebras in $C_b(X)$, introducing the concept of a twisted-sum topology and exploring its implications.

Findings

Spectrum of the sum equals disjoint union of spectra

Spectral topology is a twisted-sum topology

Applications to one-point compactification and asymptotically almost periodic functions

Abstract

Let $C_b(X)$ be the C*-algebra of bounded continuous functions on some non-compact, but locally compact Hausdorff space $X$. Moreover, let $A_0$ be some ideal and $A_1$ be some unital C*-subalgebra of $C_b(X)$. For $A_0$ and $A_1$ having trivial intersection, we show that the spectrum of their vector space sum equals the disjoint union of their individual spectra, whereas their topologies are nontrivially interwoven. Indeed, they form a so-called twisted-sum topology which we will investigate before. Within the whole framework, e.g., the one-point compactification of $X$ and the spectrum of the algebra of asymptotically almost periodic functions can be described.