Representations of dynamical systems on Banach spaces not containing $l_1$

TL;DR

This paper characterizes tame dynamical systems on topological groups through their linear representations on Banach spaces that exclude $l_1$, linking dynamical properties with Banach space geometry.

Contribution

It establishes an equivalence between tameness of G-spaces and their representation on Rosenthal Banach spaces, advancing understanding of dynamical systems and Banach space theory.

Findings

Tame G-spaces are exactly those representable on Rosenthal Banach spaces.

Provides a new characterization of tame dynamical systems via Banach space representations.

Links topological dynamics with geometric properties of Banach spaces.

Abstract

For a topological group G, we show that a compact metric G-space is tame if and only if it can be linearly represented on a separable Banach space which does not contain an isomorphic copy of $l_1$ (we call such Banach spaces, Rosenthal spaces). With this goal in mind we study tame dynamical systems and their representations on Banach spaces.