Characterization of Translation Bounded Measure Dynamical Systems and Positive Measure Dynamical Systems within the Spatial Processes

TL;DR

This paper characterizes when spatial processes are equivalent to translation bounded measure dynamical systems and positive measure dynamical systems using Gelfand theory, providing a mathematical framework for understanding these systems.

Contribution

It introduces a characterization method for spatial processes as translation bounded or positive measure dynamical systems via Gelfand theory.

Findings

Provides criteria for equivalence to translation bounded measure dynamical systems.

Provides criteria for equivalence to positive measure dynamical systems.

Uses Gelfand theory to identify elements of spatial processes with measures.

Abstract

In this paper we consider spatial processes and measure dynamical systems over locally compact Abelian groups. We characterize when a spatial processes is equivalent to a translation bounded measure dynamical systems and we characterize when a spatial processes is equivalent to a positive measure dynamical systems. The basic idea of our approach is the identification of the elements of a spatial process with translation bounded measures respectively positive measures by applying the Gelfand theory of $C^*$-algebras.