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# The Ordered $K$-theory of a Full Extension

Published:2013-07-11
Printed: Jun 2014
• Søren Eilers,
Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen, Denmark
• Gunnar Restorff,
Faculty of Science and Technology, University of Faroe Islands, Nóatún 3, FO-100 Tórshavn, Faroe Islands
• Efren Ruiz,
Department of Mathematics, University of Hawaii, Hilo, 200 W. Kawili St., Hilo, Hawaii, 96720-4091 USA
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## Abstract

Let $\mathfrak{A}$ be a $C^{*}$-algebra with real rank zero which has the stable weak cancellation property. Let $\mathfrak{I}$ be an ideal of $\mathfrak{A}$ such that $\mathfrak{I}$ is stable and satisfies the corona factorization property. We prove that $0 \to \mathfrak{I} \to \mathfrak{A} \to \mathfrak{A} / \mathfrak{I} \to 0$ 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 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.}
 Keywords: classification, extensions, graph algebras
 MSC Classifications: 46L80 - $K$-theory and operator algebras (including cyclic theory) [See also 18F25, 19Kxx, 46M20, 55Rxx, 58J22] 46L35 - Classifications of $C^*$-algebras 46L05 - General theory of $C^*$-algebras