Maximal and reduced Roe algebras of coarsely embeddable spaces

TL;DR

This paper investigates the relationship between maximal and usual Roe algebras of metric spaces, showing isomorphism in K-theory under coarse embeddability and equality under property A, using E-theoretic methods.

Contribution

It establishes conditions under which maximal and usual Roe algebras are K-theoretically equivalent or identical, extending coarse geometric understanding.

Findings

Coarse embeddability implies K-theory isomorphism between maximal and usual Roe algebras.

Property A ensures maximal and usual Roe algebras are the same.

Uses E-theoretic techniques inspired by Higson-Kasparov-Trout and Yu.

Abstract

Gong, Wang and Yu introduced a maximal, or universal, version of the Roe C*-algebra associated to a metric space. We study the relationship between this maximal Roe algebra and the usual version, in both the uniform and non-uniform cases. The main result is that if a (uniformly discrete, bounded geometry) metric space X coarsely embeds in a Hilbert space, then the canonical map between the maximal and usual (uniform) Roe algebras induces an isomorphism on K-theory. We also give a simple proof that if X has property A, then the maximal and usual (uniform) Roe algebras are the same. These two results are natural coarse-geometric analogues of certain well-known implications of a-T-menability and amenability for group C*-algebras. The techniques used are E-theoretic, building on work of Higson-Kasparov-Trout and Yu.