Purely infinite labeled graph $C^{\ast }$ -algebras

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).