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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