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

Pierre Clare, Nigel Higson, Yanli Song, Xiang Tang

Abstract: This is the first of two papers dedicated to the detailed determination of the reduced $\mathrm{C}^*$-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 $\mathrm{C}^*$-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.