The Ordered K-theory of a Full Extension

Eilers, Søren ; Restorff, Gunnar ; Ruiz, Efren

Canadian journal of mathematics, Tome 66 (2014) no. 3, pp. 596-624

Résumé

Let $\mathfrak{A}$ be a ${{C}^{*}}$ -algebra with real rank zero that has the stable weak cancellation property. Let $\Im $ be an ideal of $\mathfrak{A}$ such that $\Im $ is stable and satisfies the corona factorization property. We prove that $$0\,\to \,\Im \,\to \mathfrak{A}\,\to \,\mathfrak{A}/\Im \,\to \,0$$ is a full extension if and only if the extension is stenotic and $K$ -lexicographic. As an immediate application, we extend the classification result for graph ${{C}^{*}}$ -algebras obtained by Tomforde and the first named author to the general non-unital case. In combination with recent results by Katsura, Tomforde, West, and the first named author, our result may also be used to give a purely $K$ -theoretical description of when an essential extension of two simple and stable graph ${{C}^{*}}$ -algebras is again a graph ${{C}^{*}}$ -algebra.

Détail

DOI : 10.4153/CJM-2013-015-7

Mots-clés : 46L80, 46L35, 46L05, classification, extensions, graph algebras