Compact Subsets of the Glimm Space of a $C^{\ast }$-algebra

Abstract

If $A$ is a $\sigma$-unital $C^{\ast }$-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 $\alpha >0$ such that $K\subset \{G\in \mathrm{Glimm}(A)\mid \Vert a+G\Vert \ge \alpha \}$. This extends a 1974 result of J. Dauns to all $\sigma$-unital $C^{\ast }$-algebras. However, there is a $C^{\ast }$-algebra $A$ and a compact subset of $\mathrm{Glimm}(A)$ that is not contained in any set of the form $\{G\in \mathrm{Glimm}(A)\mid \Vert a+G\Vert \ge \alpha \}$, $a\in A$ and $\alpha >0$.

8 pages

References in corpus (1)

• Locally Inner Actions on $C_0(X)$-Algebras