This paper is about the reduced group C * -algebras of real reductive groups, and about Hilbert C * -modules over these C * -algebras. We shall do three things. First we shall apply theorems from the tempered representation theory of reductive groups to determine the structure of the reduced C * -algebra (the result has been known for some time, but it is difficult to assemble a full treatment from the existing literature). Second, we shall use the structure of the reduced C * -algebra to determine the structure of the Hilbert C * -bimodule that represents the functor of parabolic induction. Third, we shall prove that the parabolic induction bimodule admits a secondary inner product, using which we can define a functor of parabolic restriction in tempered representation theory. We shall prove in a sequel to this paper that parabolic restriction is adjoint, on both the left and the right, to parabolic induction in the context of tempered unitary Hilbert space representations.

We formulate a second adjoint theorem in the context of tempered representations of real reductive groups, and prove it in the case of SL(2, ).

Let E be a (right) Hilbert module over a C * -algebra A. If E is equipped with a left action of a second C * -algebra B, then tensor product with E gives rise to a functor from the category of Hilbert B-modules to the category of Hilbert A-modules. The purpose of this paper is to study adjunctions between functors of this sort. We shall introduce a new kind of adjunction relation, called a local adjunction, that is weaker than the standard concept from category theory. We shall give several examples, the most important of which is the functor of parabolic induction in the tempered representation theory of real reductive groups. Each local adjunction gives rise to an ordinary adjunction of functors between categories of Hilbert space representations. In this way we shall show that the parabolic induction functor has a simultaneous left and right adjoint, namely the parabolic restriction functor constructed in [CCH16].