  For the case of phase damping (pure decoherence) we investigate the extent to
which environmental traits are imprinted on an open quantum system. The
dynamics is described using the quantum channel approach. We study what the
knowledge of the channel may reveal about the nature of its underlying dynamics
and, conversely, what the dynamics tells us about how to consistently model the
environment. We find that for a Markov phase-damping channel, that is, a
channel compatible with a time-continuous Markovian evolution, the environment
may adequately be represented by a mixture of only a few coherent states. For
arbitrary Hilbert space dimension $N\geq 4$ we refine the idea of {\it quantum
phase damping}, of which we show a means of identification. Symmetry
considerations are used to identify decoherence-free subspaces of the system.
