  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 the sequel to this paper that parabolic restriction
is adjoint, on both the left and the right, to parabolic induction.
