Compact Subsets of the Glimm Space of a $C^*$-algebra

TL;DR

This paper extends a 1974 result to all sigma-unital C*-algebras, showing that compact subsets of the Glimm space can be approximated by certain norm conditions, but also presents a counterexample highlighting limitations.

Contribution

It generalizes Dauns' 1974 theorem to all sigma-unital C*-algebras and identifies limitations through a counterexample.

Findings

Extension of Dauns' result to all sigma-unital C*-algebras

Existence of compact subsets not captured by norm-based sets

Counterexample illustrating the limits of the extension

Abstract

If $A$ is a $\sigma$-unital $C^*$-algebra and $a$ is a strictly positive element of $A$ then for every compact subset $K$ of the complete regularization $\mathrm{Glimm}(A)$ of $\mathrm{Prim}(A)$ there exists $\alpha > 0$ such that $K\subset \{G\in \mathrm{Glimm}(A) \mid \|a + G\|\geq \alpha\}$. This extends a 1974 result of J. Dauns to all $\sigma$-unital $C^*$-algebras. However, there is a $C^*$-algebra $A$ and a compact subset of $\mathrm{Glimm}(A)$ that is not contained in any set of the form $\{G\in \mathrm{Glimm}(A) \mid \|a + G\|\geq \alpha\}$, $a\in A$ and $\alpha > 0$.