Skew-product dynamical systems for crossed product C*-algebras and their ergodic properties

Starting from a discrete $C^*$-dynamical system $(mathfrak{A}, heta, omega_o)$, we define and study most of the main ergodic properties of the crossed product $C^*$-dynamical system $(mathfrak{A} times_alphamathbb{Z}, Phi_{ heta, u},om_ocirc E)$, $E:mathfrak{A} times_alphamathbb{Z} ightarrowga$ being the canonical conditional expectation of $mathfrak{A} times_alphamathbb{Z}$ onto $mathfrak{A}$, provided $ainaut(ga)$ commute with the $*$-automorphism $ h$ up to a unitary $uinga$. Here, $Phi_{ heta, u}inaut(mathfrak{A} times_alphamathbb{Z})$ can be considered as the fully noncommutative generalisation of the celebrated skew-product defined by H. Anzai for the product of two tori in the classical case.