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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
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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 primitive ideal space, complete regularization
MSC Classifications: 46L05 show english descriptions General theory of $C^*$-algebras 46L05 - General theory of $C^*$-algebras
 

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