We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
$C^{\ast }$-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).
