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


Published:20121028
Printed: Mar 2014
Aldo J. Lazar,
School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel
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 $\alpha \gt 0$.