Dynamics in large scale geometry

TL;DR

This paper explores the large scale geometry of metric spaces using dynamics and operator algebras, linking dynamical behaviors to GNS representations of uniform Roe algebras.

Contribution

It establishes a novel connection between large scale dynamical phenomena and operator algebra representations via the Stone-ech boundary and uniform Roe algebras.

Findings

Characterizes Hausdorffness and T_1 properties of orbit spaces by coarse embeddability.

Shows the orbit space satisfies a localized Urysohn's lemma despite weak separation.

Provides classes of spaces where prime ideals of uniform Roe algebras are primitive.

Abstract

We investigate the large scale geometry of certain metric spaces through the lens of dynamics. Our approach establishes a close connection between large scale dynamical phenomena and operator algebras by characterizing various large scale dynamic behaviors in terms of GNS representations of the uniform Roe algebras arising from natural canonical states. Our dynamical systems are given by the Stone-\v{C}ech boundary of metric spaces together with their inverse semigroup of partial translations. This defines a space of orbits and we characterize Hausdorffness and $T_1$-ness of this space by the failure of coarse embeddability of certain metric spaces. Surprisingly, while the orbit space has very weak separation properties, we show that it satisfies a certain ''localized version'' of Urysohn's lemma. We show that the topology of the space of orbits and quasi-orbits are given by the space of irreducible representations of uniform Roe algebras and by the space of their primitive ideals, respectively. As a highlight of the theory developed herein, we provide classes of spaces such that the prime ideals of their uniform Roe algebras are primitive. This is the case for instance of spaces whose orbit space is $T_1$.