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$C^*$-ALGEBRAS OF SELF-SIMILAR ACTION OF GROUPOIDS ON ROW-FINITE DIRECTED GRAPHS

Part of: Selfadjoint operator algebras

Published online by Cambridge University Press:  08 November 2022

ISNIE YUSNITHA Open the ORCID record for ISNIE YUSNITHA [Opens in a new window]
ISNIE YUSNITHA*
Affiliation:
School of Mathematics and Applied Statistics, University of Wollongong, Wollongong NSW 2522, Australia Department of Mathematics Education, Indonesia University of Education, West Java, Indonesia e-mail: [email protected]
*
e-mail: [email protected]
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Abstract

For amenable discrete groupoids $\mathcal {G}$ and row-finite directed graphs E, let $(\mathcal {G},E)$ be a self-similar groupoid and let $C^*(\mathcal {G}, E)$ be the associated $C^*$-algebra. We introduce a weaker faithfulness condition than those in the existing literature that still guarantees that $C^*(\mathcal {G})$ embeds in $C^*(\mathcal {G}, E)$. Under this faithfulness condition, we prove a gauge-invariant uniqueness theorem.

Keywords

row-finite graphsself-similar groupoidsinjectivity conditiongauge-invariant uniqueness theorem

MSC classification

Primary: 46L05: General theory of $C^*$-algebras
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 108 , Issue 1 , August 2023 , pp. 150 - 161
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

This work is supported by a PhD scholarship of The Ministry of Education, Culture, Research and Technology of the Republic of Indonesia.

References

Exel, R. and Pardo, E., ‘Self-similar graphs: a unified treatment of Katsura and Nekrashevych ${C}^{\ast }$ -algebras’, Adv. Math. 306 (2017), 1046–1129.CrossRefGoogle Scholar
Grigorchuk, R. I., ‘Burnside’s problem on periodic groups’, Funct. Anal. Appl. 14 (1980), 41–43.CrossRefGoogle Scholar
Gupta, N. D. and Sidki, S. N., ‘On the Burnside problem for periodic groups’, Math. Z. 182 (1983), 385–388.CrossRefGoogle Scholar
Laca, M., Raeburn, I., Rammage, J. and Whittaker, M. F., ‘Equilibrium states on the Cuntz–Pimsner algebras of self-similar actions’, J. Funct. Anal. 266 (2014), 6619–6661.CrossRefGoogle Scholar
Laca, M., Raeburn, I., Rammage, J. and Whittaker, M. F., ‘Equilibrium states on operator algebras associated to self-similar actions of groupoids on graphs’, Adv. Math. 331 (2018), 268–325.CrossRefGoogle Scholar
Raeburn, I., Graph Algebras, CBMS Regional Conference Series in Mathematics, 103 (American Mathematical Society, Providence, RI, 2005).CrossRefGoogle Scholar
Sims, A., ‘Hausdorff étale groupoids and their ${C}^{\ast }$ -algebras’, in: Operator Algebras and Dynamics: Groupoids, Crossed Products and Rokhlin Dimension, Part II, Advanced Courses in Mathematics CRM Barcelona (ed. Perera, F.) (Birkhäuser, Cham, 2020).CrossRefGoogle Scholar
Sims, A., Whitehead, B. and Whittaker, M. F., ‘Twisted ${C}^{\ast }$ -algebras associated to finitely aligned higher rank graphs’, Doc. Math. (Paris) 19 (2014), 831–866.CrossRefGoogle Scholar