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Reconstructing étale groupoids from semigroups

  • Tristan Bice ORCID logo EMAIL logo and Lisa Orloff Clark
From the journal Forum Mathematicum


We unify various étale groupoid reconstruction theorems such as the following:

  1. Kumjian and Renault’s reconstruction from a groupoid C*-algebra;

  2. Exel’s reconstruction from an ample inverse semigroup;

  3. Steinberg’s reconstruction from a groupoid ring;

  4. Choi, Gardella and Thiel’s reconstruction from a groupoid Lp-algebra.

We do this by working with certain bumpy semigroups S of functions defined on an étale groupoid G. The semigroup structure of S together with the diagonal subsemigroup D then yields a natural domination relation on S. The groupoid of -ultrafilters is then isomorphic to the original groupoid G.

Communicated by Siegfried Echterhoff

Award Identifier / Grant number: 20-31529X

Award Identifier / Grant number: 67985840

Funding statement: The first author is supported by the GAČR project EXPRO 20-31529X and RVO: 67985840. The second author is supported by a Marsden Fund of the Royal Society of New Zealand.


The authors would like to thank Astrid an Huef for several conversations and contributions which helped improve the paper. The first author also thanks her for her kind hospitality while he was visiting Victoria University of Wellington in January 2020.


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Received: 2021-03-04
Revised: 2021-06-30
Published Online: 2021-09-25
Published in Print: 2021-11-01

© 2021 Walter de Gruyter GmbH, Berlin/Boston

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