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

TL;DR

This paper computes the reduced C*-algebra of real reductive groups and verifies the Connes-Kasparov conjecture, providing detailed Morita equivalence and K-theory calculations crucial for understanding their representation theory.

Contribution

It provides the first detailed proof of the Connes-Kasparov conjecture for real reductive groups, including Morita equivalence and K-theory computations.

Findings

Established Morita equivalence of the reduced C*-algebra

Computed the Connes-Kasparov morphism for these groups

Verified the conjecture for connected, linear, real reductive groups

Abstract

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.