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- W2613257862 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." @default.
- W2613257862 created "2017-05-19" @default.
- W2613257862 creator A5087210747 @default.
- W2613257862 creator A5091250545 @default.
- W2613257862 date "2017-05-10" @default.
- W2613257862 modified "2023-09-27" @default.
- W2613257862 title "Classification of Uniform Roe algebras of locally finite groups" @default.
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- W2613257862 doi "https://doi.org/10.48550/arxiv.1705.03716" @default.
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