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

The structures of state space concerning Quantum Dynamical Semigroups