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} (\underbar{K}_{\mathcal{E}} (e), \underbar{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.