K-theory for the crossed products by certain actions of Z2

We investigate the K-theory of crossed product C*-algebras by actions of Z(2). Given a 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 Z(2)-actions, where the associated obstruction homomorphisms are non-trivial. One class of examples comprises certain outer 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 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 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 Z(2)-actions with non-trivial obstruction homomorphisms. We also compute theK-theory of the crossed products associated with these universal C*-dynamical systems.