Purely infinite labeled graph $C^*$-algebras

Abstract: 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 $C^*$-algebra $C^{*(E,\mathcal{L},\mathcal{E})$} is purely infinite in the sense that every nonzero 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, $C^{(E,\mathcal{L},\mathcal{E})=C^(p_A,} s_a)$ is purely infinite in the sense of Kirchberg and R{\o}rdam 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 $C^{*(E,\mathcal{L},\mathcal{E})$} has the property (IH).