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

Aldo J. Lazar

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arXiv · 1203.3350

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

Aldo J. Lazar

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Abstract

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

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Aldo J. Lazar. 2012-03-15. Compact Subsets of the Glimm Space of a $C^*$-algebra. https://arxiv.org/abs/1203.3350

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