Reconstructing étale groupoids from semigroups
-
Tristan Bice
und
Lisa Orloff Clark
Abstract
We unify various étale groupoid reconstruction theorems such as the following:
Kumjian and Renault’s reconstruction from a groupoid C*-algebra;
Exel’s reconstruction from an ample inverse semigroup;
Steinberg’s reconstruction from a groupoid ring;
Choi, Gardella and Thiel’s reconstruction from a groupoid
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
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-2016-0072Suche 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), http://arxiv.org/abs/2010.16113. 10.1016/j.exmath.2021.07.001Suche 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.2021.106853Suche in Google Scholar
[4] T. Bice, Representing rings on ringoid bundles, preprint (2020), http://arxiv.org/abs/2012.03006. Suche in Google Scholar
[5] T. Bice, Representing semigroups on étale groupoid bundles, preprint (2020), http://arxiv.org/abs/2010.04961. 10.1017/S1446788722000064Suche in Google Scholar
[6] T. Bice, An algebraic approach to the Weyl groupoid, J. Algebra 568 (2021), 193–240. 10.1016/j.jalgebra.2020.10.010Suche in Google Scholar
[7] T. Bice, Dauns–Hofmann–Kumjian–Renault duality for Fell bundles and structured C*-algebras, preprint (2021), http://arxiv.org/abs/2104.12415. Suche 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.2018.10.031Suche in Google Scholar
[9]
J. H. Brown, L. O. Clark and A. an Huef,
Diagonal-preserving ring
[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.2015.52Suche 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/S021919971750064XSuche in Google Scholar
[12]
T. M. Carlsen, E. Ruiz, A. Sims and M. Tomforde,
Reconstruction of groupoids and
[13]
Y. Choi, E. Gardella and H. Thiel,
Rigidity results for
[14] R. Exel, Reconstructing a totally disconnected groupoid from its ample semigroup, Proc. Amer. Math. Soc. 138 (2010), no. 8, 2991–3001. 10.1090/S0002-9939-10-10346-3Suche in Google Scholar
[15] J. Goubault-Larrecq, Non-Hausdorff Topology and Domain Theory, New Math. Monogr. 22, Cambridge University, Cambridge, 2013. 10.1017/CBO9781139524438Suche in Google Scholar
[16]
A. Kumjian,
On
[17] M. V. Lawson and D. H. Lenz, Pseudogroups and their étale groupoids, Adv. Math. 244 (2013), 117–170. 10.1016/j.aim.2013.04.022Suche in Google Scholar
[18] A. N. Milgram, Multiplicative semigroups of continuous functions, Duke Math. J. 16 (1949), 377–383. 10.1215/S0012-7094-49-01638-5Suche 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.2006.09.006Suche in Google Scholar
[20]
J. Renault,
A Groupoid Approach to
[21]
J. Renault,
Cartan subalgebras in
[22] P. Resende, Étale groupoids and their quantales, Adv. Math. 208 (2007), no. 1, 147–209. 10.1016/j.aim.2006.02.004Suche in Google Scholar
[23] B. Steinberg, Diagonal-preserving isomorphisms of étale groupoid algebras, J. Algebra 518 (2019), 412–439. 10.1016/j.jalgebra.2018.10.024Suche in Google Scholar
© 2021 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- On the topological complexity of manifolds with abelian fundamental group
- Reconstructing étale groupoids from semigroups
- Fiberwise linear differential operators
- Foliations with isolated singularities on Hirzebruch surfaces
- On a stronger reconstruction notion for monoids and clones
- Cochain level May–Steenrod operations
- Commutative L-algebras and measure theory
- Rankin–Selberg integrals for principal series representations of GL(n)
- A simple proof of the generalized Leibniz rule on bounded Euclidean domains
- Construction of a class of maximal commutative subalgebras of prime Leavitt path algebras
- Seshadri constants on some Quot schemes
- Hörmander Fourier multiplier theorems with optimal Besov regularity on multi-parameter Hardy spaces
- On spectral and non-spectral problem for the planar self-similar measures with four element digit sets
- C-minimal topological groups
- Cevian properties in ideal lattices of Abelian ℓ-groups
- Study of twisted Bargmann transform via Bargmann transform
Artikel in diesem Heft
- Frontmatter
- On the topological complexity of manifolds with abelian fundamental group
- Reconstructing étale groupoids from semigroups
- Fiberwise linear differential operators
- Foliations with isolated singularities on Hirzebruch surfaces
- On a stronger reconstruction notion for monoids and clones
- Cochain level May–Steenrod operations
- Commutative L-algebras and measure theory
- Rankin–Selberg integrals for principal series representations of GL(n)
- A simple proof of the generalized Leibniz rule on bounded Euclidean domains
- Construction of a class of maximal commutative subalgebras of prime Leavitt path algebras
- Seshadri constants on some Quot schemes
- Hörmander Fourier multiplier theorems with optimal Besov regularity on multi-parameter Hardy spaces
- On spectral and non-spectral problem for the planar self-similar measures with four element digit sets
- C-minimal topological groups
- Cevian properties in ideal lattices of Abelian ℓ-groups
- Study of twisted Bargmann transform via Bargmann transform
