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This is the first of two papers dedicated to the computation of the reducedC*-algebra of a connected, linear, real reductive group up to Moritaequivalence, and the verification of the Connes-Kasparov conjecture for thesegroups. These results were originally announced by Antony Wassermann in 1987.In Part I we shall give details of the C*-algebraic Morita equivalence, andthen compute the Connes-Kasparov morphism subject to some results in temperedrepresentation theory that we shall prove in Part II using tools from DavidVogan's classification of the tempered dual.

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