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Purely infinite labeled graph 
$C^{\ast }$-algebras
Published online by Cambridge University Press: 04 December 2017
Abstract
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In this paper, we consider pure infiniteness of generalized Cuntz–Krieger algebras associated to labeled spaces 
$(E,{\mathcal{L}},{\mathcal{E}})$. It is shown that a 
$C^{\ast }$-algebra 
$C^{\ast }(E,{\mathcal{L}},{\mathcal{E}})$ is purely infinite in the sense that every non-zero hereditary subalgebra contains an infinite projection (we call this property (IH)) if 
$(E,{\mathcal{L}},{\mathcal{E}})$ is disagreeable and every vertex connects to a loop. We also prove that under the condition analogous to (K) for usual graphs, 
$C^{\ast }(E,{\mathcal{L}},{\mathcal{E}})=C^{\ast }(p_{A},s_{a})$ is purely infinite in the sense of Kirchberg and Rørdam if and only if every generating projection 
$p_{A}$, 
$A\in {\mathcal{E}}$, is properly infinite, and also if and only if every quotient of 
$C^{\ast }(E,{\mathcal{L}},{\mathcal{E}})$ has property (IH).
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