Measurable bundles of $$C^*$$ C ∗ -dynamical systems and its applications

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 \(L_0\).