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The Ordered K-theory of a Full Extension

Published online by Cambridge University Press:  20 November 2018

Søren Eilers ,
Gunnar Restorff  and
Efren Ruiz
Søren Eilers
Affiliation:
Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen, Denmark. e-mail: eilers@math.ku.dk
Gunnar Restorff
Affiliation:
Faculty of Science and Technology, University of Faroe Islands, Nóatún 3, FO-100 Tórshavn, Faroe Islands. e-mail: gunnarr@setur.fo
Efren Ruiz
Affiliation:
Department of Mathematics, University of Hawaii, Hilo, 200 W. Kawili St., Hilo, Hawaii, 96720-4091, USA. e-mail: ruize@hawaii.edu
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Abstract

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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.

Keywords

46L8046L3546L05classificationextensionsgraph algebras
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 66 , Issue 3 , 01 June 2014 , pp. 596 - 624
Copyright
Copyright © Canadian Mathematical Society 2014

References

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