On the Connes-Kasparov isomorphism, I: The reduced C*-algebra of a real reductive group and the K-theory of the tempered dual

This is the first of two papers dedicated to the computation of the 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. These results were originally announced by Antony Wassermann in 1987. In Part I we shall give details of the C*-algebraic Morita equivalence, and then compute the Connes-Kasparov morphism subject to some results in tempered representation theory that we shall prove in Part II using tools from David Vogan’s classification of the tempered dual.