# Gelfand-type theorems for dynamical Banach modules

## TL;DR

This paper extends classical Gelfand-type theorems to dynamical systems on Banach bundles, providing new representation results for weighted Koopman operators on Banach modules.

## Contribution

It introduces Gelfand-type theorems for dynamics on Banach bundles and their associated weighted Koopman representations, broadening the scope of classical functional analysis results.

## Key findings

- Established Gelfand-type theorems for topological Banach bundles
- Developed representation theorems for measurable Banach bundles
- Extended Koopman linearization to Banach modules

## Abstract

The representation theorems of Gelfand and Kakutani for commutative C*-algebras and AM- and AL-spaces are the basis for the Koopman linearization of topological and measure-preserving dynamical systems. In this article we prove versions of these results for dynamics on topological and measurable Banach bundles and the corresponding weighted Koopman representations on Banach modules.