Classification of Uniform Roe algebras of locally finite groups

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 $C_u^{\ast }(\Gamma )$ and $C_u^{\ast }(\Lambda )$ are $\ast$-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 $C^{\ast }$-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 $\ast$-isomorphic. Finally, we give a summary of $C^{\ast }$-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 ${\ell }^{\infty }(\Gamma ){⋊}_r\Gamma$ is locally finite-dimensional.

20 pages

References in corpus (1)

• Continuous orbit equivalence rigidity