The Global Glimm Property for C*-algebras of topological dimension zero

TL;DR

This paper characterizes when C*-algebras of topological dimension zero have the Global Glimm Property, linking it to being nowhere scattered, and shows such algebras are pure if they also have finite nuclear dimension.

Contribution

It provides a complete characterization of the Global Glimm Property for topological dimension zero C*-algebras, solving the Global Glimm Problem in this context.

Findings

C*-algebras with topological dimension zero have the Global Glimm Property iff they are nowhere scattered.

Nowhere scattered C*-algebras with finite nuclear dimension and topological dimension zero are pure.

The paper establishes a necessary and sufficient condition for the Global Glimm Property in this class.

Abstract

We show that a C*-algebra with topological dimension zero has the Global Glimm Property (every hereditary subalgebra contains an almost full nilpotent element) if and only if it is nowhere scattered (no hereditary subalgebra admits a finite-dimensional representation). This solves the Global Glimm Problem in this setting. It follows that nowhere scattered C*-algebras with finite nuclear dimension and topological dimension zero are pure.