Abstract
In the present paper we investigate \(L_0\)-valued states and Markov operators on \( C^*\)-algebras over \(L_0\). Here, \(L_0=L_0(\Omega )\) is the algebra of equivalence classes of complex measurable functions on \((\Omega ,\Sigma ,\mu )\). In particular, we give representations for \(L_0\)-valued states and Markov operators on \(C^*\)-algebras over \(L_0\), respectively, as measurable bundles of states and Markov operators. Moreover, we apply the obtained representations to study certain ergodic properties of \( C^*\)-dynamical systems over \(L_0\).
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Acknowledgments
The authors are grateful to Professor Vladimir Chilin and an anonymous referee for their valuable comments and remarks on improving the paper. This work has been supported by the IIUM Grant EDW B 11-185-0663 and the MOHE grants FRGS11-022-0170, FRGS13-071-0312, ERGS13-024-0057. The second named author (F.M.) acknowledges the Junior Associate scheme of the Abdus Salam International Centre for Theoretical Physics, Trieste, Italy.
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Ganiev, I., Mukhamedov, F. Measurable bundles of \(C^*\)-dynamical systems and its applications. Positivity 18, 687–702 (2014). https://doi.org/10.1007/s11117-013-0270-4
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DOI: https://doi.org/10.1007/s11117-013-0270-4