The ideal structure of Cuntz–Krieger algebras

Abstract

We construct a universal Cuntz–Krieger algebra ${\cal {AO}}_A$, which is isomorphic to the usual Cuntz–Krieger algebra ${\cal O}_A$ when $A$ satisfies condition $(I)$ of Cuntz and Krieger. The Cuntz classification of ideals in ${\cal O}_A$ when $A$ satisfies condition $(II)$ extends to a classification of the gauge invariant ideals in ${\cal {AO}}_A$. We use this to describe the topology on the primitive ideal space of ${\cal {AO}}_A$.