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Compact Subsets of the Glimm Space of a C*-algebra
Published online by Cambridge University Press: 20 November 2018
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 $\alpha \,>\,0$.
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- Copyright © Canadian Mathematical Society 2014