Automorphisms of Cuntz–Krieger algebras

Abstract

We prove that the natural homomorphism from Kirchberg’s ideal-related $KK$-theory, $K{K}_{\mathcal{E}}(e,e^{\prime })$, with one specified ideal, into ${\mathrm{Hom}}_{\Lambda }({\underline{\text{K}}}_{\mathcal{E}}(e),{\underline{\text{K}}}_{\mathcal{E}}(e^{\prime }))$ is an isomorphism for all extensions $e$ and $e^{\prime }$ of separable, nuclear $C^{\ast }$-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.