A groupoid picture of the Elek algebras
DOI:
https://doi.org/10.7146/math.scand.a-122419Abstract
We reformulate a construction by Gábor Elek, which associates $C^{\ast} $-algebras with uniformly recurrent subgroups, in the language of groupoid $C^{\ast} $-algebras. This allows us to simplify several proofs from the original paper and add the converse direction to Elek's characterisation of nuclearity, showing that his sufficient condition is in fact necessary. We furthermore relate our groupoids to the dynamics of the group acting on its uniformly recurrent subgroup.
References
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Borys, C., The Furstenberg boundary of a groupoid, eprint arXiv:1904.10062 [math.OA], 2019.
Elek, G., Uniformly recurrent subgroups and simple $C^*$-algebras, J. Funct. Anal. 274 (2018), no. 6, 1657–1689. https://doi.org/10.1016/j.jfa.2018.01.004
Glasner, E. and Weiss, B., Uniformly recurrent subgroups, in “Recent trends in ergodic theory and dynamical systems”, Contemp. Math., vol. 631, Amer. Math. Soc., Providence, RI, 2015, pp. 63–75. https://doi.org/10.1090/conm/631/12596
Kennedy, M., An intrinsic characterization of $C^\ast $-simplicity, eprint arXiv:1509.01870 [math.OA], 2015.
Renault, J., A groupoid approach to $C^\ast $-algebras, Lecture Notes in Mathematics, vol. 793, Springer, Berlin, 1980.
Sims, A., Étale groupoids and their $C^\ast $-algebras, eprint arXiv:1710.10897 [math.OA], 2017.
Sims, A. and Williams, D. P., Amenability for Fell bundles over groupoids, Illinois J. Math. 57 (2013), no. 2, 429–444.
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Published
2021-08-31
How to Cite
Borys, C. (2021). A groupoid picture of the Elek algebras. MATHEMATICA SCANDINAVICA, 127(2), 185–208. https://doi.org/10.7146/math.scand.a-122419
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