In 1987, Antony Wassermann  announced a result of the structure of reduced $C^∗$-algebra of a connected, linear real reductive group, up to Morita equivalence, and the verification of the Connes-Kasparov conjecture for these groups. In this talk, I will close a gap in the literature by providing the remaining details concerning the computation of the reduced $C^∗$-algebra and discuss details of the $C^∗$-algebraic Morita equivalence. In addition, I will also review the construction of the Connes-Kasparov morphism. The tool we used in the computation comes from David Vogan’s theory of minimal K-types. This is a joint work with Pierre Clare, Nigel Higson and Xiang Tang.