Ideal structures and rigidity of uniform Roe algebras

TL;DR

This paper investigates the rigidity of uniform Roe algebras by examining their ideal structures, establishing that isomorphic uniform Roe algebras imply coarse equivalence of the underlying metric spaces.

Contribution

It demonstrates that isomorphism of uniform Roe algebras for spaces with bounded geometry implies coarse equivalence, revealing a rigidity property.

Findings

Isomorphic uniform Roe algebras imply coarse equivalence of metric spaces.

The ideal structure of uniform Roe algebras encodes coarse geometric information.

Rigidity results connect algebraic isomorphisms to geometric equivalences.

Abstract

In this paper, we study the rigidity of uniform Roe algebras via the ideal structures. We showed that for given metric spaces X and Y with bounded geometry, if their uniform Roe algebras are isomorphic, then they are coarse equivalent.