General uniform Roe algebra rigidity

TL;DR

This paper extends the rigidity results of uniform Roe algebras to all uniformly locally finite coarse spaces, establishing conditions under which algebraic isomorphisms imply coarse equivalences of the underlying spaces.

Contribution

It generalizes known rigidity results to broader classes of coarse spaces and characterizes embeddings and Cartan subalgebras within uniform Roe algebras.

Findings

Isomorphism implies coarse equivalence under certain conditions

Spaces with property A are bijectively coarsely equivalent

Characterization of embeddings via underlying spaces

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.