Journal of Noncommutative Geometry

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Volume 12, Issue 1, 2018, pp. 217–254
DOI: 10.4171/JNCG/275

Published online: 2018-03-23

Automorphisms of Cuntz–Krieger algebras

Søren Eilers[1], Gunnar Restorff[2] and Efren Ruiz[3]

(1) University of Copenhagen, Denmark
(2) University of the Faroe Islands, Tórshavn, Faroe Islands
(3) University of Hawaii, Hilo, USA

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.

Keywords: KK-theory, UCT, Cuntz–Krieger algebras, automorphisms

Eilers Søren, Restorff Gunnar, Ruiz Efren: Automorphisms of Cuntz–Krieger algebras. J. Noncommut. Geom. 12 (2018), 217-254. doi: 10.4171/JNCG/275