location:  Publications → journals → CMB
Abstract view

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

Published:2012-10-28
Printed: Mar 2014
• Aldo J. Lazar,
School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel
Features coming soon:
Citations   (via CrossRef) Tools: Search Google Scholar:
 Format: LaTeX MathJax PDF

## 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 $\alpha \gt 0$ such that $K\subset \{G\in \mathrm{Glimm}(A) \mid \Vert a + G\Vert \geq \alpha\}$. This extends a result of J. Dauns to all $\sigma$-unital $C^*$-algebras. However, there are a $C^*$-algebra $A$ and a compact subset of $\mathrm{Glimm}(A)$ that is not contained in any set of the form $\{G\in \mathrm{Glimm}(A) \mid \Vert a + G\Vert \geq \alpha\}$, $a\in A$ and $\alpha \gt 0$.
 Keywords: primitive ideal space, complete regularization
 MSC Classifications: 46L05 - General theory of $C^*$-algebras