JournalsjncgVol. 10, No. 4pp. 1559–1587

KK-theory for the crossed products by certain actions of Z2\mathbb Z^2

  • Selçuk Barlak

    University of Southern Denmark, Odense, Denmark
$K$-theory for the crossed products by certain actions of $\mathbb Z^2$ cover
Download PDF

Abstract

We investigate the KK-theory of crossed product CC^*-algebras by actions of Z2\mathbb Z^2. Given a Z2\mathbb Z^2-action, we associate to it a homomorphism between certain subquotients of the KK-theory of the underlying CC^*-algebra, which we call the obstruction homomorphism. This homomorphism together with the KK-theory of the underlying algebra and the induced action in KK-theory determine the KK-theory of the associated crossed product CC^*-algebra up to group extension problems. A concrete description of this obstruction homomorphism is provided as well. We give examples of Z2\mathbb Z^2-actions, where the associated obstruction homomorphisms are non-trivial. One class of examples comprises certain outer Z2\mathbb Z^2-actions on Kirchberg algebras, which act trivially on KKKK-theory. This relies on a classification result by Izumi and Matui. A second class of examples consists of certain pointwise inner Z2\mathbb Z^2-actions. One instance is given as a natural action on the group CC^*-algebra of the discrete Heisenberg group. We also compute the KK-theory of the corresponding crossed product. A general and concrete construction yields various examples of pointwise inner Z2\mathbb Z^2-actions on amalgamated free product CC^*-algebras with non-trivial obstruction homomorphisms. Among these, there are actions that are universal, in a suitable sense, for pointwise inner Z2\mathbb Z^2-actions with non-trivial obstruction homomorphisms. We also compute the KK-theory of the crossed products associated with these universal CC^*-dynamical systems.

Cite this article

Selçuk Barlak, KK-theory for the crossed products by certain actions of Z2\mathbb Z^2. J. Noncommut. Geom. 10 (2016), no. 4, pp. 1559–1587

DOI 10.4171/JNCG/266