Continuous orbit equivalence of groupoid actions

Xiangqi Qiang; Xiangqi Qiang

doi:10.3934/math.2026548

AIMS Mathematics

2026, Volume 11, Issue 5: 13287-13303. doi: 10.3934/math.2026548

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Research article

Continuous orbit equivalence of groupoid actions

Xiangqi Qiang ^,

School of Science, Jiangsu University of Science and Technology, Zhenjiang 212100, China

Received: 02 February 2026 Revised: 30 April 2026 Accepted: 06 May 2026 Published: 13 May 2026
MSC : 37B05, 46L05, 46L35

In this paper, I studied continuous actions of étale groupoids on compact Hausdorff spaces and the associated operator algebras. I introduced notions of conjugacy and continuous orbit equivalence for such groupoid actions and characterized them in terms of the corresponding transformation groupoids and their reduced $ C^* $-algebras. In particular, I proved that two topologically free actions are continuously orbit equivalent if and only if their associated transformation groupoid $ C^* $-algebras are isomorphic, and if and only if there exists a $ C^* $-algebra isomorphism preserving the canonical Cartan subalgebras between the corresponding reduced $ C^* $-algebras of these transformation groupoids.
- groupoid action,
- transformation groupoid,
- continuous orbit equivalence,
- conjugacy,
- groupoid $ C^{*} $-algebra
Citation: Xiangqi Qiang. Continuous orbit equivalence of groupoid actions[J]. AIMS Mathematics, 2026, 11(5): 13287-13303. doi: 10.3934/math.2026548

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Abstract

In this paper, I studied continuous actions of étale groupoids on compact Hausdorff spaces and the associated operator algebras. I introduced notions of conjugacy and continuous orbit equivalence for such groupoid actions and characterized them in terms of the corresponding transformation groupoids and their reduced $ C^* $-algebras. In particular, I proved that two topologically free actions are continuously orbit equivalent if and only if their associated transformation groupoid $ C^* $-algebras are isomorphic, and if and only if there exists a $ C^* $-algebra isomorphism preserving the canonical Cartan subalgebras between the corresponding reduced $ C^* $-algebras of these transformation groupoids.

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