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.

To a measured space carrying two group actions, we associate a Hilbert C * -module in a way that generalises Rieffel's construction of induction modules. This construction is then applied to describe the P -series of a semisimple Lie group. We provide several realisations of this module, corresponding to the classical pictures for the P -series. We also characterise a class of bounded operators on the module which satisfy some commutation relation, and interpret the result as a generic irreducibility theorem. Finally, we establish the convergence of standard intertwining integrals on a dense subset of this module.2000 Mathematics Subject Classification. 46L08, 22D25, 22E46.

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].