On a Rokhlin property for abelian group actions on C$^*$-algebras

In this article, we study the so-called abelian Rokhlin property for actions of locally compact, abelian groups on C$^*$-algebras. We propose a unifying framework for obtaining various duality results related to this property. The abelian Rokhlin property coincides with the known Rokhlin property for actions by the reals (i.e., flows), but is not identical to the known Rokhlin property in general. The main duality result we obtain is a generalisation of a duality for flows proved by Kishimoto in the case of Kirchberg algebras. We consider also a slight weakening of the abelian Rokhlin property, which allows us to show that all traces on the crossed product C$^*$-algebra are canonically induced from invariant traces on the the coefficient C$^*$-algebra.