Compact Subsets of the Glimm Space of a C*-algebra. Issue 1 (14th March 2014)

• Record Type: Journal Article
• Authors: Lazar, Aldo J.
• Is Part Of: Canadian mathematical bulletin =. Volume 57:Issue 1(2014)
• Page Start: 90
• Page End: 96
• Publication Date: 2014-03-14
• Subjects: 46L05; primitive ideal space -- complete regularization; Mathematics -- Periodicals; Mathematics; Periodicals; 510.5
• DOI: 10.4153/CMB-2012-038-2
• Languages: English
• ISSNs: 0008-4395

Abstract

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 Glimm $(A)$ of Prim $(A)$ there exists $\alpha \, >\, 0$ such that $K\, \subset \, \{G\, \in \, \text{Glimm(}A\text{)}\, \text{ }\!\!|\!\!\text{ }\, \left\| a\, +\, G \right\|\, \ge \, \alpha \}$ . This extends a result of J. Dauns to all $\sigma $ -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 $\{G\, \in \, \text{Glimm(}A\text{)}\, \text{ }\!\!|\!\!\text{ }\, \left\| a+\, G \right\|\, \ge \, \alpha \}, \, a\in \, A$ and $\alpha \, >\, 0$ .