Compact Subsets of the Glimm Space of a C*-algebra. Issue 1 (14th March 2014) - Lazar, Aldo J.
Abstract: If $A$ is a $σ $ -unital ${C^{*}}$ -algebra and $a$ is a strictly positive element of $A$, then for every compact subset $K$ of the complete regularization Glimm $(A)$ of Prim $(A)$ there exists $α \, >\, 0$ such that $K\, \subset \, \{G\, \in \, \text{Glimm(}A\text{)}\, \text{ }\!\!|\!\!\text{ }\, \left\| a\, +\, G \right\|\, \ge \, α \}$ . This extends a result of J. Dauns to all $σ $ -unital ${C^{*}}$ -algebras. However, there exist a ${C^{*}}$ -algebra $A$ and a compact subset of Glimm $(A)$ that is not contained in any set of the form $\{G\, \in \, \text{Glimm(}A\text{)}\, \text{ }\!\!|\!\!\text{ }\, \left\| a+\, G \right\|\, \ge \, α \}, \, a\in \, A$ and $α \, >\, 0$ .