A Mackey embedding for reduced $C^*$-algebras of real reductive groups

The purpose of this paper is to construct an embedding of the $C^*$-algebra of the Cartan motion group of a real reductive group $G$ into the reduced $C^*$-algebra of $G$ itself. The embedding has a number of applications: we shall use it to characterize the Mackey bijection from the tempered dual of $G$ into the unitary dual of the motion group; to characterize the continuous field of reduced group $C^*$-algebras arising from the contraction of $G$ to its Cartan motion group; and to characterize the Connes-Kasparov assembly map in operator $K$-theory. The results presented here continue and complete a project that was begun several years ago by the last two authors, who considered the case of complex groups. In the real case, detailed information from the theory of $R$-groups is used in the construction.