Fragmentability and representations of flows

TL;DR

This paper investigates weak star continuous representations of semigroup actions on dual Banach spaces, introducing Asplund functions and exploring fragmentability to unify dynamical properties and generalize classical results.

Contribution

It introduces Asplund functions, links fragmentability with flow properties, and extends known results on minimal subsets and orbitwise Kadec actions.

Findings

Weakly almost periodic flows admit many reflexive representations.

Fragmentability acts as a generalized form of equicontinuity.

Linear G-actions often have coinciding weak and strong topologies on minimal sets.

Abstract

Our aim is to study weak star continuous representations of semigroup actions into the duals of ``good'' (e.g., reflexive and Asplund) Banach spaces. This approach leads to flow analogs of Eberlein and Radon-Nikodym compacta and a new class of functions (Asplund functions) which intimately is connected with Asplund representations and includes the class of weakly almost periodic functions. We show that a flow is weakly almost periodic iff it admits sufficiently many reflexive representations. One of the main technical tools in this paper is the concept of fragmentability (which actually comes from Namioka and Phelps) and widespreadly used in topological aspects of Banach space theory. We explore fragmentability as ``a generalized equicontinuity'' of flows. This unified approach allows us to obtain several dynamical applications. We generalize and strengthen some results of Akin-Auslander-Berg, Shtern, Veech-Troallic-Auslander and Hansel-Troallic. We establish that frequently, for linear G-actions, weak and strong topologies coincide on, not necessarily closed, G-minimal subsets. For instance such actions are ``orbitwise Kadec``.