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Equivalence and stable isomorphism of groupoids, and diagonal-preserving stable isomorphisms of graph $ C^*$-algebras and Leavitt path algebras

Authors: Toke Meier Carlsen, Efren Ruiz and Aidan Sims
Journal: Proc. Amer. Math. Soc. 145 (2017), 1581-1592
MSC (2010): Primary 46L05; Secondary 16S99
Published electronically: October 27, 2016
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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.

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Toke Meier Carlsen
Affiliation: Department of Science and Technology, University of the Faroe Islands Nóatún 3, FO-100 Tórshavn, the Faroe Islands

Efren Ruiz
Affiliation: Department of Mathematics, University of Hawaii, Hilo, 200 W. Kawili Street, Hilo, Hawaii 96720-4091

Aidan Sims
Affiliation: School of Mathematics and Applied Statistics, University of Wollongong, NSW 2522, Australia

Keywords: Stabilisation, stable isomorphism, groupoid equivalence, Kakutani equivalence, graph algebra, Leavitt path algebra
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
Article copyright: © Copyright 2016 American Mathematical Society