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

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^*-algebr...

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Veröffentlicht in:	Proceedings of the American Mathematical Society 2017-04, Vol.145 (4), p.1581
Hauptverfasser:	Toke Meier Carlsen, Efren Ruiz, Aidan Sims
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Sprache:	eng
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creator	Toke Meier Carlsen Efren Ruiz Aidan Sims
description	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.
doi_str_mv	10.1090/proc/13321
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title	Equivalence and stable isomorphism of groupoids, and diagonal-preserving stable isomorphisms of graph C^-algebras and Leavitt path algebras
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