  Let E be a (right) Hilbert C*-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
a previous paper.
