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By Bekka's theorem the group C*-algebra of an amenable group $G$ isresidually finite dimensional (RFD) if and only if $G$ is maximally almostperiodic (MAP). We generalize this result in two directions of dynamicalflavour. Firstly, we provide a sufficient condition for the RFD property of theC*-algebra of an amenable \'etale groupoid. Secondly, we characterize RFDproperty for crossed products by amenable actions of discrete groups onC*-algebras. The characterisation can be formulated in various terms, such asprimitive ideals, (pure) states and approximations of representations, and canbe viewed as a dynamical version of Exel-Loring characterization of RFDC*-algebras. This leads among other consequences to a characterization of whena semidirect product by an amenable group has RFD full C*-algebra. As byproductof our methods we also characterize the property FD of Lubotzky and Shalom forsemidirect products by amenable groups and obtain characterizations of theproperties MAP and RF for general semidirect products of groups. The latterdescriptions allow us to obtain the properties MAP, RF, RFD and FD for variousexamples.

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