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A correspondence between inverse subsemigroups, open wide subgroupoids and cartan intermediate C*-subalgebras

Part of: Topological and differentiable algebraic systems Semigroups Selfadjoint operator algebras

Published online by Cambridge University Press:  07 October 2022

Fuyuta Komura Open the ORCID record for Fuyuta Komura [Opens in a new window]
Fuyuta Komura*
Affiliation:
Graduate School of Science and Technology, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan ([email protected])
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Abstract

For a given inverse semigroup action on a topological space, one can associate an étale groupoid. We prove that there exists a correspondence between the certain subsemigroups and the open wide subgroupoids in case that the action is strongly tight. Combining with the recent result of Brown et al., we obtain a correspondence between the certain subsemigroups of an inverse semigroup and the Cartan intermediate subalgebras of a groupoid C*-algebra.

Keywords

C*-algebrasétale groupoidsinverse semigroups

MSC classification

Primary: 20M18: Inverse semigroups 22A22: Topological groupoids (including differentiable and Lie groupoids) 46L05: General theory of $C^*$-algebras
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 65 , Issue 3 , August 2022 , pp. 861 - 879
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Copyright
Copyright © The Author(s), 2022. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

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References

Brown, J. H., Exel, R., Fuller, A. H., Pitts, D. R. and Reznikoff, S. A., Intermediate C*-algebras of Cartan embeddings, Proc. Amer. Math. Soc. Ser. B 8 (2021), 27–41.CrossRefGoogle Scholar
Exel, R., Inverse semigroups and combinatorial C*-algebras, Bull. Brazilian Math. Soc., New Seri. 39 (2007), 191–313.CrossRefGoogle Scholar
Exel, R. and Pardo, E., The tight groupoid of an inverse semigroup, Semi. Forum 92(1) (2016), 274–303.CrossRefGoogle Scholar
Lawson, M. V., Inverse semigroups: the theory of partial symmetries (World Scientific, 1998).CrossRefGoogle Scholar
Lawson, M. V., The polycyclic monoids $P_n$ and the Thompson groups $V_n,\, 1$, Communi. Algebra 35(12) (2007), 4068–4087.CrossRefGoogle Scholar
Lawson, M. V., Compactable semilattices, Semi. Forum 81(1) (2010), 187–199.CrossRefGoogle Scholar
Matsnev, D. and Resende, P., Étale groupoids as germ groupoids and their base extensions, Proc. Edinburgh Math. Soc. 53(3) (2010), 765–785.CrossRefGoogle Scholar
Paterson, A., Groupoids. Progress in Mathematics. (Birkhäuser, Boston, 2012).Google Scholar
Renault, J., A groupoid approach to C*-Algebras, Lecture Notes in Mathematics (Springer-Verlag, 1980).CrossRefGoogle Scholar
Renault, J., Cartan subalgebras in C*-algebras, Irish Math. Soc. Bull. 61 (2008), 29–63.CrossRefGoogle Scholar
Sims, A., Hausdorff étale groupoids and their C*-algebras. Operator Algebras and Dynamics: Groupoids, Crossed Products, and Rokhlin Dimension (2020).Google Scholar
Steinberg, B., A groupoid approach to discrete inverse semigroup algebras, Adv. Math. 223(2) (2010), 689–727.CrossRefGoogle Scholar