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

Lazar, Aldo J.

Canadian mathematical bulletin, Tome 57 (2014) no. 1, pp. 90-96

Résumé

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 $\alpha \,>\,0$ .

Détail

DOI : 10.4153/CMB-2012-038-2

Mots-clés : 46L05, primitive ideal space, complete regularization