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 provide a sufficient condition for the RFD property of the C*-algebra of an amenable \'etale groupoid. Secondly, we characterize RFD property for crossed products by amenable actions of discrete groups on C*-algebras. The characterisation can be formulated in various terms, such as primitive ideals, (pure) states and approximations of representations, and can be viewed as a dynamical version of Exel-Loring characterization of RFD C*-algebras. As byproduct of our methods we also 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. The latter descriptions allow us to obtain the properties MAP, RF, RFD and FD for various examples.