Automorphisms of Cuntz–Krieger algebras

  • Søren Eilers

    University of Copenhagen, Denmark
  • Gunnar Restorff

    University of the Faroe Islands, Tórshavn, Faroe Islands
  • Efren Ruiz

    University of Hawaii, Hilo, USA
Automorphisms of Cuntz–Krieger algebras cover
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Abstract

We prove that the natural homomorphism from Kirchberg’s ideal-related KKKK-theory, KKE(e,e)KK_\mathcal E(e, e'), with one specified ideal, into HomΛ(\ushortKE(e),\ushortKE(e))\mathrm{Hom}_{\Lambda} (\ushort{K}_{\mathcal{E}} (e), \ushort{K}_{\mathcal{E}} (e')) is an isomorphism for all extensions ee and ee' of separable, nuclear CC^{*}-algebras in the bootstrap category N\mathcal{N} with the KK-groups of the associated cyclic six term exact sequence being finitely generated, having zero exponential map and with the K1K_{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.

Cite this article

Søren Eilers, Gunnar Restorff, Efren Ruiz, Automorphisms of Cuntz–Krieger algebras. J. Noncommut. Geom. 12 (2018), no. 1, pp. 217–254

DOI 10.4171/JNCG/275