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1. CJM 2013 (vol 66 pp. 596)
The Ordered $K$-theory of a Full Extension 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 Categories:46L80, 46L35, 46L05 |
2. CJM 2006 (vol 58 pp. 1268)
Gauge-Invariant Ideals in the $C^*$-Algebras of Finitely Aligned Higher-Rank Graphs We produce a complete description of the lattice of gauge-invariant
ideals in $C^*(\Lambda)$ for a finitely aligned $k$-graph
$\Lambda$. We provide a condition on $\Lambda$ under which every ideal
is gauge-invariant. We give conditions on $\Lambda$ under which
$C^*(\Lambda)$ satisfies the hypotheses of the Kirchberg--Phillips
classification theorem.
Keywords:Graphs as categories, graph algebra, $C^*$-algebra Category:46L05 |