Skew-product dynamical systems for crossed product $C^{\ast }$-algebras and their ergodic properties

Abstract

Starting from a discrete $C^{\ast }$-dynamical system $(\mathfrak{A},\theta ,{\omega }_o)$, we define and study most of the main ergodic properties of the crossed product $C^{\ast }$-dynamical system $(\mathfrak{A}\rtimes_α\mathbb{Z}, Φ_{θ, u},\om_o\circ E)$, $E:\mathfrak{A}\rtimes_α\mathbb{Z}\rightarrow\ga$ being the canonical conditional expectation of $\mathfrak{A}{⋊}_{\alpha }\mathbb{Z}$ onto $\mathfrak{A}$, provided $\a\in\aut(\ga)$ commute with the $\ast$-automorphism $þ$ up tu a unitary $u\in\ga$. Here, $Φ_{θ, u}\in\aut(\mathfrak{A}\rtimes_α\mathbb{Z})$ 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.

To appear in Journal of Mathematical Analysis and Applications