Equivalence and stable isomorphism of groupoids, and diagonal-preserving stable isomorphisms of graph $C^*$-algebras and Leavitt path algebras
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- by Toke Meier Carlsen, Efren Ruiz and Aidan Sims PDF
- Proc. Amer. Math. Soc. 145 (2017), 1581-1592 Request permission
Abstract:
We prove that ample groupoids with $\sigma$-compact unit spaces are equivalent if and only if they are stably isomorphic in an appropriate sense, and relate this to Matui’s notion of Kakutani equivalence. We use this result to show that diagonal-preserving stable isomorphisms of graph $C^*$-algebras or Leavitt path algebras give rise to isomorphisms of the groupoids of the associated stabilised graphs. We deduce that the Leavitt path algebras $L_{\mathbb {Z}}(E_2)$ and $L_{\mathbb {Z}}(E_{2-})$ are not stably $^*$-isomorphic.References
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Additional Information
- Toke Meier Carlsen
- Affiliation: Department of Science and Technology, University of the Faroe Islands Nóatún 3, FO-100 Tórshavn, the Faroe Islands
- MR Author ID: 685180
- ORCID: 0000-0002-7981-7130
- Email: toke.carlsen@gmail.com
- Efren Ruiz
- Affiliation: Department of Mathematics, University of Hawaii, Hilo, 200 W. Kawili Street, Hilo, Hawaii 96720-4091
- MR Author ID: 817213
- Email: ruize@hawaii.edu
- Aidan Sims
- Affiliation: School of Mathematics and Applied Statistics, University of Wollongong, NSW 2522, Australia
- MR Author ID: 671497
- Email: asims@uow.edu.au
- Received by editor(s): February 8, 2016
- Received by editor(s) in revised form: May 23, 2016
- Published electronically: October 27, 2016
- Additional Notes: This work was initiated, and much of it completed, while all three authors were attending the research program Classification of operator algebras: complexity, rigidity, and dynamics at the Mittag-Leffler Institute, January–April 2016. This research was supported by Australian Research Council grant DP150101598 and by a grant from the Simons Foundation (#279369 to the second author). The authors thank Dana Williams for helpful email correspondence.
- Communicated by: Adrian Ioana
- © Copyright 2016 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 145 (2017), 1581-1592
- MSC (2010): Primary 46L05; Secondary 16S99
- DOI: https://doi.org/10.1090/proc/13321
- MathSciNet review: 3601549