Rigidity for geometric ideals in uniform Roe algebras

TL;DR

This paper studies the rigidity of geometric ideals in uniform Roe algebras, showing that stable isomorphism implies coarse equivalence of the underlying spaces, with discussions on ghostly ideals and open questions.

Contribution

It establishes a rigidity result linking stable isomorphism of geometric ideals to coarse equivalence of spaces, advancing understanding of uniform Roe algebra structures.

Findings

Stable isomorphism implies coarse equivalence of associated spaces.

Characterization of geometric ideals via coarse structures.

Discussion on ghostly ideals and open problems.

Abstract

In this paper, we investigate the rigidity problems for geometric ideals in uniform Roe algebras associated to discrete metric spaces of bounded geometry. These ideals were introduced by Chen and Wang, and can be fully characterised in terms of ideals in the associated coarse structures. Our main result is that if two geometric ideals in uniform Roe algebras are stably isomorphic, then the coarse spaces associated to these ideals are coarsely equivalent. We also discuss the case of ghostly ideals and pose some open questions.