Non-commutative skew-product extension dynamical systems

Starting from a uniquely ergodic action of a locally compact group $G$ on a compact space $X_0$, we consider non-commutative skew-product extensions of the dynamics, on the crossed product $C(X_0)\rtimes_\alpha\mathbb{Z}$, through a $1$-cocycle of $G$ in $\mathbb{T}$, with $\alpha$ commuting with the given dynamics. We first prove that any such two skew-product extensions are conjugate if and only if the corresponding cocycles are cohomologous. We then study unique ergodicity and unique ergodicity w.r.t. the fixed-point subalgebra by characterizing both in terms of the cocycle assigning the dynamics. The set of all invariant states is also determined: it is affinely homeomorphic with $\mathcal{P}(\mathbb{T})$, the Borel probability measures on the one-dimensional torus $\mathbb{T}$, as long as the system is not uniquely ergodic. Finally, we show that unique ergodicity w.r.t. the fixed-point subalgebra of a skew-product extension amounts to the uniqueness of an invariant conditional expectation onto the fixed-point subalgebra