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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 classification, extensions, graph algebras
MSC Classifications: 46L80, 46L35, 46L05 show english descriptions $K$-theory and operator algebras (including cyclic theory) [See also 18F25, 19Kxx, 46M20, 55Rxx, 58J22]
Classifications of $C^*$-algebras
General theory of $C^*$-algebras
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
 

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