C*-rigidity of bounded geometry metric spaces

TL;DR

This paper establishes a deep connection between the algebraic structure of Roe algebras and the coarse geometric properties of bounded geometry metric spaces, showing that algebraic isomorphisms imply coarse equivalences.

Contribution

It proves that isomorphic Roe algebras of uniformly locally finite metric spaces imply the spaces are coarsely equivalent, revealing a form of C*-rigidity.

Findings

Isomorphic Roe algebras imply coarse equivalence of spaces

Outer automorphism group of Roe algebra corresponds to coarse automorphisms

Provides new insights into the structure of metric spaces via operator algebras

Abstract

We prove that uniformly locally finite metric spaces with isomorphic Roe algebras must be coarsely equivalent. As an application, we also prove that the outer automorphism group of the Roe algebra of a metric space of bounded geometry is canonically isomorphic to the group of coarse equivalences of the space up to closeness.