It is known that the group $C^*$-algebra of a compact group is an AF-algebra. I want to know if given a non-compact locally compact group $G$, does there exist conditions on $G$ which imply that the (full or reduced) group $C^*$-algebra of $G$ is an AF-algebra?
Also, are there any known examples of non-compact locally compact groups whose group $C^*$-algebra is an AF-algebra, and the group $C^*$-algebra has been computed? Examples where the group is non-discrete would be of most interest.