Automorphisms of Cuntz-Krieger algebras

TL;DR

This paper establishes an isomorphism between KK-theory and homomorphisms for certain C*-algebra extensions, leading to classification results for automorphisms of stabilized Cuntz-Krieger algebras with one ideal.

Contribution

It proves a new isomorphism in KK-theory for a class of C*-algebras and classifies automorphisms of stabilized Cuntz-Krieger algebras with one ideal.

Findings

Isomorphism between KK-theory and homomorphisms for specific extensions

Classification of automorphisms modulo asymptotic unitary equivalence

Applicable to certain graph algebras

Abstract

We prove that the natural homomorphism from Kirchberg's ideal-related KK-theory, KKE(e, e'), with one specified ideal, into Hom_{\Lambda} (\underline{K}_{E} (e), \underline{K}_{E} (e')) is an isomorphism for all extensions e and e' of separable, nuclear C*-algebras in the bootstrap category 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.