# Journal of Operator Theory

Volume 90, Issue 1, Summer 2023 pp. 263-310.

Stably finite extensions of $C^*$-algebras of
rank-two graphs

**Authors**:
Astrid an Huef (1),
Abraham C.S. Ng (2), and Aidan Sims (3)

**Author institution:**
(1) School of Mathematics and Statistics, Victoria University of Wellington, P.O. Box 600, Wellington 6140, New Zealand

(2) School of Mathematics and Applied Statistics,
University of Wollongong,
NSW 2522, Australia

(3) School of Mathematics and Applied Statistics,
University of Wollongong,
NSW 2522, Australia

**Summary: ** We study stable finiteness of extensions of $2$-graph
$C^*$-algebras determined by saturated hereditary sets of vertices. We use
two iterations of the Pimsner-Voiculescu sequence to calculate the map in
$K$-theory induced by the inclusion of a hereditary subgraph into the larger
$2$-graph it lives in. We then apply a theorem of Spielberg about stable
finiteness of extensions to provide a sufficient condition for the
$C^*$-algebra of the larger $2$-graph to be stably finite. We illustrate our
results with examples.

**DOI: **http://dx.doi.org/10.7900/jot.2021Nov29.2376

**Keywords: ** Higher-rank graph, $k$-graph, stably finite $C^*$-algebra,
$K$-theory, extension, Pimsner-Voiculescu sequence.

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