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

TL;DR

This paper explores $L_0$-valued states and Markov operators on $C^*$-algebras over $L_0$, providing representations as measurable bundles and applying these to analyze ergodic properties of such dynamical systems.

Contribution

It introduces a novel representation framework for $L_0$-valued states and Markov operators on $C^*$-algebras over $L_0$, enabling new analysis of their ergodic behavior.

Findings

Representations of $L_0$-valued states as measurable bundles

Representations of Markov operators as measurable bundles

Application to ergodic properties of $C^*$-dynamical systems

Abstract

In the present paper we investigate $L_0$-valued states and Markov operators on $ C^*$-algebras over $L_0$. In particular, we give representations for $L_0$-valued state 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 $L_0$.