General Uniform Roe algebra rigidity

Abstract

We generalize all known results on rigidity of uniform Roe algebras to the setting of arbitrary uniformly locally finite coarse spaces. For instance, we show that isomorphism between uniform Roe algebras of uniformly locally finite coarse spaces whose uniform Roe algebras contain only compact ghost projections implies that the base spaces are coarsely equivalent. Moreover, if one of the spaces has property A, then the base spaces are bijectively coarsely equivalent. We also provide a characterization for the existence of an embedding onto hereditary subalgebra in terms of the underlying spaces. As an application, we partially answer a question of White and Willett about Cartan subalgebras of uniform Roe algebras.Résumé. -Nous généralisons tous les résultats connus sur la rigidité des algèbres de Roe uniformes au cas des espaces grossiers uniformément localement finis arbitraires. Nous montrons que l'isomorphisme entre les algèbres de Roe uniformes des espaces grossiers uniformément localement finis dont les algèbres de Roe uniformes ne contiennent que des projections fantômes compactes implique que les espaces de base sont grossièrement équivalents. De plus, si l'un des espaces a la propriété A, alors les espaces de base sont bijectivement grossièrement équivalents. Nous fournissons également une caractérisation de l'existence d'un plongement sur les sous-algèbres héréditaires en termes de sous-espaces. Comme application, nous répondons partiellement à une question de White et Willett sur les sous-algèbres de Cartan des algèbres de Roe uniformes.

Coarse quotients of metric spaces and embeddings of uniform Roe algebras

“…Also, the map f is clearly injective, and, by [5,Lemma 5.3], it is also coarse. 6 We now show that f W X ! Z is co-coarse.…”

“…Precisely, if .C u .X /; B/ satisfies (2), it is not known whether B is automatically co-separable. By [6,Corollary 6.3], this is indeed the case if X has property A. The importance of Roe Cartan pairs to our goals lies in the following theorem:…”

“…N/ is automatically coseparable. However, it has been recently shown that this is indeed the case if Y has property A [6,Corollary 6.3].…”