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The study demonstrates the Global Glimm Property in C-algebras, implying insights into their structure and representation.
We show that a ‐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 ‐algebras with finite nuclear dimension and topological dimension zero are pure.
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Ng et al. (2026) studied this question.
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