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

Published online by Cambridge University Press:  20 November 2018

Aldo J. Lazar
Aldo J. Lazar*
Affiliation:
School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel e-mail: aldo@post.tau.ac.il
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Abstract

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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 $\alpha \,>\,0$.

Keywords

46L05primitive ideal spacecomplete regularization
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 57 , Issue 1 , 14 March 2014 , pp. 90 - 96
Copyright
Copyright © Canadian Mathematical Society 2014

References

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