We start by introducing C*-algebras associated with locally compact groups. Next, the theory of Hilbert modules, C*-correspondences, crossed-product algebras, and Morita equivalence are discussed. This is followed with applications to Mackey’s theory of induction of representations and Howe’s theory of theta correspondence. Next, K-theory and (equivariant) KK-theory are introduced. Connections between isolated points in the unitary dual and generators of the K-theory of C*-algebras of liminal groups are discussed.

This chapter is an introduction to the geometric perspective of tempered representations of connected reductive Lie groups via Dirac induction and the Connes–Kasparov isomorphism. We start with discussing Dirac operators in spin geometry. We then recall some classical Dirac operators on homogeneous spaces arising in representation theory. Next, we introduce the theory of higher index to be used as a language to formulate the Connes–Kasparov conjecture. In Section 3.4, we introduce Dirac induction, the Connes–Kasparov conjecture, and survey a couple of ways for proving it. The final section is devoted to the harmonic-analytic approach to the Connes–Kasparov isomorphism illustrated by the example of connected semisimple groups.