We investigate the -theory of crossed product -algebras by actions of . Given a -action, we associate to it a homomorphism between certain subquotients of the -theory of the underlying -algebra, which we call the obstruction homomorphism. This homomorphism together with the -theory of the underlying algebra and the induced action in -theory determine the -theory of the associated crossed product -algebra up to group extension problems. A concrete description of this obstruction homomorphism is provided as well. We give examples of -actions, where the associated obstruction homomorphisms are non-trivial. One class of examples comprises certain outer -actions on Kirchberg algebras, which act trivially on -theory. This relies on a classification result by Izumi and Matui. A second class of examples consists of certain pointwise inner -actions. One instance is given as a natural action on the group -algebra of the discrete Heisenberg group. We also compute the -theory of the corresponding crossed product. A general and concrete construction yields various examples of pointwise inner -actions on amalgamated free product -algebras with non-trivial obstruction homomorphisms. Among these, there are actions that are universal, in a suitable sense, for pointwise inner -actions with non-trivial obstruction homomorphisms. We also compute the -theory of the crossed products associated with these universal -dynamical systems.
Cite this article
Selçuk Barlak, -theory for the crossed products by certain actions of . J. Noncommut. Geom. 10 (2016), no. 4, pp. 1559–1587DOI 10.4171/JNCG/266