  We show that some abstract results on propagation of fixed points for
completely positive maps on $C^*$-algebras provide a natural approach to unify
recent Noether type theorems on the equivalence of symmetries with conservation
laws for dynamical systems of Markov processes, of quantum operations, and of
quantum stochastic maps. In addition, we obtain some new Noether type theorems,
provide examples and counter-examples, and extend most of the existing results
with characterisations in terms of dual infinitesimal generators of the
corresponding strongly continuous one-parameter semigroups.
