Purely infinite labeled graph $C^{\ast }$-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 $C^{\ast }$-algebra $C^{\ast }(E,\mathcal{L},\mathcal{E})$ 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, $C^{\ast }(E,\mathcal{L},\mathcal{E})=C^{\ast }(p_A,s_a)$ 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 the property (IH).

32 pages