Non-commutative skew-product extension dynamical systems

Starting from a uniquely ergodic action of a locally compact group G on a compact space X0, we consider non-commutative skew-product extensions of the dynamics, on the crossed product C(X0) α Z, through a 1-cocycle of G in T, with α commuting with the given dynamics. We first prove that any two such skew-product extensions are conjugate if and only if the corresponding cocycles are cohomologous. We then study unique ergodicity and unique ergodicity with respect to 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 P(T), the Borel probability measures on the one-dimensional torus T, as long as the system is not uniquely ergodic. Finally, we show that unique ergodicity with respect to the fixed-point subalgebra of a skew-product extension amounts to the uniqueness of an invariant conditional expectation onto the fixed-point subalgebra.