  Each semigroup describing the time evolution of an open quantum system on a
finite dimensional Hilbert space is related to a special structure of this
space. It is shown how the space can be decomposed into subspaces: One is
related to decay, orthogonal subspaces support the stationary states.
Specialities where the complete positivity of evolutions is actually needed for
analysis, mainly for evolution of coherence, are highlighted. Decompositions
are done the same way for evolutions in discrete as in continuous time, but
evolutions may show differences, only for discrete semigroups there may appear
cases of sudden decay and of perpetual oscillation. Concluding the analysis we
identify the relation of the state space structure to the processes of Decay,
Decoherence, Dissipation and Dephasing.
