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Skew-product dynamical systems for crossed product \(C^\ast \)-algebras and their ergodic properties. (English) Zbl 1480.46082

Summary: 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 \((\mathfrak{A}\rtimes_\alpha \mathbb{Z},\Phi_{\theta,u},\omega_o\circ E)\), \(E:\mathfrak{A}\rtimes_\alpha \mathbb{Z}\to\mathfrak{A}\) being the canonical conditional expectation of \(\mathfrak{A}\rtimes_\alpha\mathbb{Z}\) onto \(\mathfrak{A}\), provided \(\alpha\in\mathrm{Aut}(\mathfrak{A})\) commutes with the \(\ast\)-automorphism \(\theta\) up to a unitary \(u\in\mathfrak{A}\). Here, \(\Phi_{\theta,u}\in\mathrm{Aut}(\mathfrak{A}\rtimes_\alpha\mathbb{Z})\) can be considered as the fully noncommutative generalisation of the celebrated skew-product defined by H. Anzai [Osaka Math. J. 3, 83–99 (1951; Zbl 0043.11203)] for the product of two tori in the classical case.

MSC:

46L55 Noncommutative dynamical systems
37A30 Ergodic theorems, spectral theory, Markov operators
37A55 Dynamical systems and the theory of \(C^*\)-algebras

Citations:

Zbl 0043.11203
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