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 detailed determination of the reduced C*-algebra of a connected, linear, real reductive group up to Morita equivalence, and a new and very explicit proof of the Connes–Kasparov conjecture for these groups using representation theory. In this part we shall give details of the C*-algebraic Morita equivalence and then explain how the Connes–Kasparov morphism in operator K-theory may be computed using what we call the Matching Theorem, which is a purely representation-theoretic result. We shall prove our Matching Theorem in the sequel, and indeed go further by giving a simple, direct construction of the components of the tempered dual that have non-trivial K-theory using David Vogan’s approach to the classification of the tempered dual.