# $K$-theory for the crossed products by certain actions of $\mathbb Z^2$

### Selçuk Barlak

University of Southern Denmark, Odense, Denmark

## Abstract

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

## Cite this article

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

DOI 10.4171/JNCG/266