Let A \mathcal {A} be an AF C ∗ {\text {AF}}\;{C^\ast } -algebra, and let δ \delta be a closed ∗ \ast -derivation which annihilates the maximal abelian subalgebra C \mathcal {C} of diagonal elements of A \mathcal {A} . Then we show that δ \delta generates an approximately inner C ∗ {C^\ast } -dynamics on A \mathcal {A} , and that δ \delta is a commutative ∗ \ast -derivation. Any two closed ∗ \ast -derivations vanishing on C \mathcal {C} are shown to be strongly commuting. More generally, if δ \delta is a semiderivation on A \mathcal {A} which vanishes on C \mathcal {C} , we prove that δ \delta is a generator of a semigroup of strongly positive contractions of A \mathcal {A} .