RFD property for groupoid C*-algebras of amenable groupoids and for crossed products by amenable actions

Abstract

By Bekka's theorem the group C*-algebra of an amenable group $G$ is residually finite dimensional (RFD) if and only if $G$ is maximally almost periodic (MAP). We generalize this result in two directions of dynamical flavour. Firstly, we completely characterize the RFD property for crossed products by amenable actions of discrete groups on C*-algebras in terms of the action. The characterisation can be formulated in various terms, such as primitive ideals, (pure) states and approximations of representations, and the latter can be viewed as a dynamical version of Exel-Loring characterization of RFD C*-algebras. %The result leads among other consequences to a characterization of when a semidirect product by an amenable group has RFD full C*-algebra. As byproduct of our methods we characterize the property FD of Lubotzky and Shalom for semidirect products by amenable groups and obtain characterizations of the properties MAP and RF for general semidirect products of groups. These descriptions allow us to obtain the properties MAP, RF, RFD and FD for various new examples and generalize some results of Lubotzky and Shalom. Secondly, as another generalization of Bekka's theorem, we provide a sufficient condition and a necessary condition for the C*-algebra of an amenable \'etale groupoid to be RFD.