Automorphisms of Cuntz–Krieger algebras

Abstract

We prove that the natural homomorphism from Kirchberg’s ideal-related $KK$-theory, $KK\_\mathcal E(e, e')$, with one specified ideal, into $\mathrm{Hom}{\Lambda} (\ushort{K}{\mathcal{E}} (e), \ushort{K}{\mathcal{E}} (e'))$ is an isomorphism for all extensions $e$ and $e'$ of separable, nuclear $C^{\*}$-algebras in the bootstrap category $\mathcal{N}$ with the $K$-groups of the associated cyclic six term exact sequence being finitely generated, having zero exponential map and with the $K{1}$-groups of the quotients being free abelian groups. This class includes all Cuntz–Krieger algebras with exactly one non-trivial ideal. Combining our results with the results of Kirchberg, we classify automorphisms of the stabilized purely infinite Cuntz–Krieger algebras with exactly one non-trivial ideal modulo asymptotically unitary equivalence. We also get a classification result modulo approximately unitary equivalence. The results in this paper also apply to certain graph algebras.