Functoriality and Weyl Groupoids of Ample C*-Diagonal Pairs
Abstract
We develop a functorial framework for ample C*-diagonal pairs and their Weyl groupoids. The main contribution is a partial functoriality theorem in the untwisted setting: from a diagonal-preserving *-homomorphism whose restriction to the diagonal is an isomorphism, we construct an open subgroupoid ℋΦ⊆𝒢B and a continuous, open, injective groupoid homomorphism ρΦ: ℋΦ→𝒢A, together with the induced homeomorphism on unit spaces. We prove that this construction is well-defined and extends to a contravariant functor into a category of partial groupoid morphisms, thereby giving a concrete partial Weyl-groupoid functor. We record consequences for expectation-compatible ideals, faithful conditional expectations, and quotient reductions, and formulate questions concerning the interaction with dynamical comparison. We also record tensor-product permanence results, including the product description of Weyl groupoids and the relevant diagonal-dimension estimate. Finally, in the untwisted principal case, we prove a reconstruction/rigidity theorem showing that the underlying Weyl groupoid determines the diagonal pair up to isomorphism. Results imported from Bönicke--Li, Li--Liao--Winter, Renault, Kumjian, and Kopsacheilis--Winter are used explicitly as background or supporting permanence results rather than claimed as new contributions.
Related
- Global and local triviality for extensions of ample groupoids: an algebraic approach
- Groupoid strict comparison for Ample Groupoids: Fiberwise Supramenability and Topological Amenability
- The Almost Finite--Purely Infinite Dichotomy for Minimal Amenable Ample Groupoids
- Discretisation in algebraic K-theory of independent ample groupoids and their Steinberg algebras
- Functoriality of logarithmic Hochschild homology of log smooth pairs
BibTeX
@misc{jabbari2026functoriality,
title = {{Functoriality and Weyl Groupoids of Ample C*-Diagonal Pairs}},
author = {Ali Jabbari},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2605.25627},
eprint = {2605.25627},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.25627},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Functoriality and Weyl Groupoids of Ample C*-Diagonal Pairs
AU - Ali Jabbari
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2605.25627
UR - https://arxiv.org/abs/2605.25627
AB - We develop a functorial framework for ample C^*-diagonal pairs and their Weyl groupoids. The main contribution is a partial functoriality theorem in the untwisted setting: from a diagonal-preserving *-homomorphism whose restriction to the diagonal is an isomorphism, we construct an open subgroupoid ℋ_Φ⊆𝒢_B and a continuous, open, injective groupoid homomorphism ρ_Φ: ℋ_Φ→𝒢_A, together with the induced homeomorphism on unit spaces. We prove that this construction is well-defined and extends to a contravariant functor into a category of partial groupoid morphisms, thereby giving a concrete partial Weyl-groupoid functor. We record consequences for expectation-compatible ideals, faithful conditional expectations, and quotient reductions, and formulate questions concerning the interaction with dynamical comparison. We also record tensor-product permanence results, including the product description of Weyl groupoids and the relevant diagonal-dimension estimate. Finally, in the untwisted principal case, we prove a reconstruction/rigidity theorem showing that the underlying Weyl groupoid determines the diagonal pair up to isomorphism. Results imported from Bönicke--Li, Li--Liao--Winter, Renault, Kumjian, and Kopsacheilis--Winter are used explicitly as background or supporting permanence results rather than claimed as new contributions.
ER -