Journal of Operator Theory

Volume 80, Issue 1, Summer 2018 pp. 25-46.

Classification of uniform Roe algebras of locally finite groups

Authors: Kang Li (1) and Hung-Chang Liao (2)
Author institution: (1) Mathematisches Institut der WWU Munster, Munster, 48149, Deutschland
(2) Mathematisches Institut der WWU Munster, Munster, 48149, Deutschland

Summary: We show that for two countable locally finite groups $\Gamma$ and $\Lambda$, the associated uniform Roe algebras $C^*_\mathrm u(\Gamma)$ and $C^*_\mathrm u(\Lambda)$ are $*$-isomorphic if and only if their $K_0$ groups are isomorphic as ordered abelian groups with units. Along the way we obtain a rigidity result: two countable locally finite groups are bijectively coarsely equivalent if and only if the associated uniform Roe algebras are $*$-isomorphic. We also show that a (not necessarily countable) discrete group $\Gamma$ is locally finite if and only if the associated uniform Roe algebra $\ell^\infty(\Gamma)\rtimes_\mathrm r \Gamma$ is locally finite-dimensional.

DOI: http://dx.doi.org/10.7900/jot.2017may23.2163
Keywords: uniform Roe algebras, classification of $C^∗$-algebras, coarse geometry

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