Skip to content
Accessible Unlicensed Requires Authentication Published by De Gruyter September 25, 2021

Reconstructing étale groupoids from semigroups

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

Abstract

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


Funding source: Grantová Agentura České Republiky

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.

Acknowledgements

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.

References

[1] P. Ara, J. Bosa, R. Hazrat and A. Sims, Reconstruction of graded groupoids from graded Steinberg algebras, Forum Math. 29 (2017), no. 5, 1023–1037. 10.1515/forum-2016-0072Search in Google Scholar

[2] B. Armstrong, L. O. Clark, A. an Huef, M. Jones and Y.-F. Lin, Filtering germs: Groupoids associated to inverse semigroups, preprint (2020), . 10.1016/j.exmath.2021.07.001Search in Google Scholar

[3] B. Armstrong, L. O. Clark, K. Courtney, Y.-F. Lin, K. McCormick and J. Ramagge, Twisted Steinberg algebras, J. Pure Appl. Algebra 226 (2022), no. 3, Paper No. 106853. 10.1016/j.jpaa.2021.106853Search in Google Scholar

[4] T. Bice, Representing rings on ringoid bundles, preprint (2020), . Search in Google Scholar

[5] T. Bice, Representing semigroups on étale groupoid bundles, preprint (2020), . Search in Google Scholar

[6] T. Bice, An algebraic approach to the Weyl groupoid, J. Algebra 568 (2021), 193–240. 10.1016/j.jalgebra.2020.10.010Search in Google Scholar

[7] T. Bice, Dauns–Hofmann–Kumjian–Renault duality for Fell bundles and structured C*-algebras, preprint (2021), . Search in Google Scholar

[8] T. Bice and C. Starling, General non-commutative locally compact locally Hausdorff Stone duality, Adv. Math. 341 (2019), 40–91. 10.1016/j.aim.2018.10.031Search in Google Scholar

[9] J. H. Brown, L. O. Clark and A. an Huef, Diagonal-preserving ring *-isomorphisms of Leavitt path algebras, J. Pure Appl. Algebra 221 (2017), no. 10, 2458–2481. 10.1016/j.jpaa.2016.12.032Search in Google Scholar

[10] N. Brownlowe, T. M. Carlsen and M. F. Whittaker, Graph algebras and orbit equivalence, Ergodic Theory Dynam. Systems 37 (2017), no. 2, 389–417. 10.1017/etds.2015.52Search in Google Scholar

[11] T. M. Carlsen and J. Rout, Diagonal-preserving graded isomorphisms of Steinberg algebras, Commun. Contemp. Math. 20 (2018), no. 6, Article ID 1750064. 10.1142/S021919971750064XSearch in Google Scholar

[12] T. M. Carlsen, E. Ruiz, A. Sims and M. Tomforde, Reconstruction of groupoids and C*-rigidity of dynamical systems, Adv. Math. 390 (2021), Paper No. 107923. 10.1016/j.aim.2021.107923Search in Google Scholar

[13] Y. Choi, E. Gardella and H. Thiel, Rigidity results for lp-operator algebras and applications, preprint (2019), . 10.29007/s94sSearch in Google Scholar

[14] R. Exel, Reconstructing a totally disconnected groupoid from its ample semigroup, Proc. Amer. Math. Soc. 138 (2010), no. 8, 2991–3001. 10.1090/S0002-9939-10-10346-3Search in Google Scholar

[15] J. Goubault-Larrecq, Non-Hausdorff Topology and Domain Theory, New Math. Monogr. 22, Cambridge University, Cambridge, 2013. 10.1017/CBO9781139524438Search in Google Scholar

[16] A. Kumjian, On C-diagonals, Canad. J. Math. 38 (1986), no. 4, 969–1008. 10.4153/CJM-1986-048-0Search in Google Scholar

[17] M. V. Lawson and D. H. Lenz, Pseudogroups and their étale groupoids, Adv. Math. 244 (2013), 117–170. 10.1016/j.aim.2013.04.022Search in Google Scholar

[18] A. N. Milgram, Multiplicative semigroups of continuous functions, Duke Math. J. 16 (1949), 377–383. 10.1215/S0012-7094-49-01638-5Search in Google Scholar

[19] J. Mrčun, On duality between étale groupoids and Hopf algebroids, J. Pure Appl. Algebra 210 (2007), no. 1, 267–282. 10.1016/j.jpaa.2006.09.006Search in Google Scholar

[20] J. Renault, A Groupoid Approach to C-Algebras, Lecture Notes in Math. 793, Springer, Berlin, 1980. 10.1007/BFb0091072Search in Google Scholar

[21] J. Renault, Cartan subalgebras in C*-algebras, Irish Math. Soc. Bull. 61 (2008), 29–63. 10.33232/BIMS.0061.29.63Search in Google Scholar

[22] P. Resende, Étale groupoids and their quantales, Adv. Math. 208 (2007), no. 1, 147–209. 10.1016/j.aim.2006.02.004Search in Google Scholar

[23] B. Steinberg, Diagonal-preserving isomorphisms of étale groupoid algebras, J. Algebra 518 (2019), 412–439. 10.1016/j.jalgebra.2018.10.024Search in Google Scholar

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