  In this note we recall some recent progress in understanding the
representation theory of *-algebras over rings C = R(i) where R is ordered and
i^2 = -1. The representation spaces are modules over auxiliary *-algebras with
inner products taking values in this auxiliary *-algebra. The ring ordering
allows to implement positivity requirements for the inner products. Then the
representations are required to be compatible with the inner product. Moreover,
one can add the notion of symmetry in form of Hopf algebra actions. For all
these notions of representations there is a well-established Morita theory
which we review. The core of each version of Morita theory is the corresponding
Picard groupoid for which we give tools to compute and determine both the
orbits and the isotropy groups.
