  We solve, mainly by counterexamples, many natural questions regarding maximal
commutative subalgebras invariant under CP-maps or semigroups of CP-maps on a
von Neumann algebra. In particular, we discuss the structure of the generators
of norm continuous semigroups on B(G) leaving a maximal commutative subalgebra
invariant and show that there exists Markov CP-semigroups on M_d without
invariant maximal commutative subalgebras for any d>2.
