  Anzai skew-products are shown to be uniquely ergodic with respect to the
fixed-point subalgebra if and only if there is a unique conditional expectation
onto such a subalgebra which is invariant under the dynamics. For the
particular case of skew-products, this solves a question raised by B. Abadie
and K. Dykema in the wider context of $C^*$-dynamical systems.
