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

Abstract

We investigate the $K$-theory of crossed product $C^{\ast }$-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 $C^{\ast }$-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 $C^{\ast }$-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 $C^{\ast }$-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 $C^{\ast }$-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 $C^{\ast }$-dynamical systems.