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).
Categories: math.OA