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

Journal article, 2026

We show that a (Formula presented.) -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 (Formula presented.) -algebras with finite nuclear dimension and topological dimension zero are pure.