  We analyze the situation of a local quantum field theory with constraints,
both indexed by the same set of space-time regions. In particular we find
``weak'' Haag-Kastler axioms which will ensure that the final constrained
theory satisfies the usual Haag-Kastler axioms. Gupta-Bleuler electromagnetism
is developed in detail as an example of a theory which satisfies the ``weak''
Haag-Kastler axioms but not the usual ones. This analysis is done by pure
C*-algebraic means without employing any indefinite metric representations, and
we obtain the same physical algebra and positive energy representation for it
than by the usual means. The price for avoiding the indefinite metric, is the
use of nonregular representations and complex valued test functions. We also
exhibit the precise connection with the usual indefinite metric representation.
We conclude the analysis by comparing the final physical algebra produced by a
system of local constrainings with the one obtained from a single global
constraining and also consider the issue of reduction by stages. For the usual
spectral condition on the generators of the translation group, we also find a
``weak'' version, and show that the Gupta-Bleuler example satisfies it.
