Dynamics in large scale geometry

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 $T_1$.