  In the context of deformation quantization, there exist various procedures to
deal with the quantization of a reduced space M_red. We shall be concerned here
mainly with the classical Marsden-Weinstein reduction, assuming that we have a
proper action of a Lie group G on a Poisson manifold M, with a moment map J for
which zero is a regular value. For the quantization, we follow [BHW] (with a
simplified approach) and build a star product *_red on M_red from a strongly
invariant star product * on M. The new questions which are addressed in this
paper concern the existence of natural *-involutions on the reduced quantum
algebra and the representation theory for such a reduced *-algebra. We assume
that * is Hermitian and we show that the choice of a formal series of smooth
densities on the embedded coisotropic submanifold C = J^{-1}(0), with some
equivariance property, defines a *-involution for *_red on the reduced space.
Looking into the question whether the corresponding *-involution is the complex
conjugation (which is a *-involution in the Marsden-Weinstein context) yields a
new notion of quantized unimodular class. We introduce a left *-submodule and a
right *_red-submodule C^\infty_cf(C)[[lambda]] of C^\infty(C)[[lambda]]; we
define on it a C^\infty(M_red)[[lambda]]-valued inner product and we establish
that this gives a strong Morita equivalence bimodule between
C^\infty(M_red)[[lambda]] and the finite rank operators on
C^\inftycf(C)[[lambda]]. The crucial point is here to show the complete
positivity of the inner product. We obtain a Rieffel induction functor from the
strongly non-degenerate *-representations of (C^\infty(M_red)[[lambda]], *_red)
on pre-Hilbert right D-modules to those of (C^\infty(M)[[lambda]], *), for any
auxiliary coefficient *-algebra D over \mathbb{C}[[lambda]].
