Classification of Uniform Roe algebras of locally finite groups

TL;DR

This paper classifies uniform Roe algebras of countable locally finite groups using K-theory, establishing a new classification framework for non-separable, non-simple C*-algebras and linking algebraic properties to geometric group features.

Contribution

It provides the first classification result for a class of non-separable unital C*-algebras, connecting algebraic isomorphisms with K-theoretic invariants and coarse geometric equivalences.

Findings

Uniform Roe algebras are classified by their K_0 groups as ordered abelian groups with units.

Two countable locally finite groups are coarsely equivalent iff their uniform Roe algebras are *-isomorphic.

Characterization of locally finite groups via their associated uniform Roe algebras.

Abstract

We study the uniform Roe algebras associated to locally finite groups. We show that for two countable locally finite groups $\Gamma$ and $\Lambda$, the associated uniform Roe algebras $C^*_u(\Gamma)$ and $C^*_u(\Lambda)$ are $*$-isomorphic if and only if their $K_0$ groups are isomorphic as ordered abelian groups with units. This can be seen as a non-separable non-simple analogue of the Glimm-Elliott classification of UHF algebras. To the best of our knowledge, this is the first classification result for a class of non-separable unital $C^*$-algebras. Along the way we also obtain a rigidity result: two countable locally finite groups are bijectively coarsely equivalent if and only if the associated uniform Roe algebras are $*$-isomorphic. Finally, we give a summary of $C^*$-algebraic characterizations for (not necessarily countable) locally finite discrete groups in terms of their uniform Roe algebras. In particular, we show that a discrete group $\Gamma$ is locally finite if and only if the associated uniform Roe algebra $\ell^\infty(\Gamma)\rtimes_r \Gamma$ is locally finite-dimensional.