Title: 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).