The rigidity problem for uniform Roe algebras

TL;DR

This paper proves that if two uniformly locally finite metric spaces have isomorphic uniform Roe algebras, then they are bijectively coarsely equivalent, establishing a rigidity result in coarse geometry.

Contribution

The paper establishes the rigidity of uniform Roe algebras by showing isomorphism implies bijective coarse equivalence of the underlying spaces.

Findings

Isomorphic uniform Roe algebras imply bijective coarse equivalence

Rigidity result for uniformly locally finite metric spaces

Advances understanding of the relationship between algebraic and geometric properties

Abstract

We solve the rigidity problem for uniform Roe algebras, by showing that two uniformly locally finite metric spaces with isomorphic uniform Roe algebras are bijectively coarsely equivalent.