Saturation and elementary equivalence of C*-algebras

TL;DR

This paper investigates the saturation properties of various classes of C*-algebras, extending known results to non-sigma-unital coronas and relating saturation to topological features of underlying spaces.

Contribution

It extends the understanding of saturation in C*-algebras by analyzing non-sigma-unital coronas and linking saturation of abelian C*-algebras to topological properties.

Findings

Some coronas of non-sigma-unital C*-algebras are countably degree-1 saturated

Saturation of C(X) relates to topological properties of X

Saturation of CL(X) reflects properties of the space X

Abstract

We study the saturation properties of several classes of $C^*$-algebras. Saturation has been shown by Farah and Hart to unify the proofs of several properties of coronas of $\sigma$-unital $C^*$-algebras; we extend their results by showing that some coronas of non-$\sigma$-unital $C^*$-algebras are countably degree-$1$ saturated. We then relate saturation of the abelian $C^*$-algebra $C(X)$, where $X$ is $0$-dimensional, to topological properties of $X$, particularly the saturation of $CL(X)$.