  It has been suggested that star products in phase-space quantization may be
augmented to describe additional, classical effects. That proposal is examined
critically here. Two known star products that introduce classical effects are:
the generalized Husimi product of coarse-grained quantization, and a
non-Hermitian damped star product for the harmonic oscillator. Following these
examples, we consider products related by transition differential operators to
the classic Moyal star product. We restrict to Hermitian star products,
avoiding problems already pointed out for the original damped product. It is
shown, however, that with such star products, augmented quantization is
impossible, since an appropriate classical limit does not result.
  For a more complete study, we then also consider generalized, or local,
transition operators, that depend on the local phase-space coordinates, as well
as their derivatives. In this framework, one example of possible physical
interest is constructed. Because of its limited validity and complicated form,
however, it cannot be concluded that augmented quantization with local
transition operators is practical.
