C*-irreducible regular inclusions, Galois correspondence and aperiodicity
Abstract
We characterise C*-irreducible regular C*-inclusions using a number of different conditions considered by different authors. In particular, we show that all C*-irreducible regular inclusions A⊆ B are modelled by outer Fell bundles (Bg)g∈ G over discrete groups with a simple unit fibre A=B1. In this case, we prove a bijection between intermediate C*-algebras A⊆ C ⊆ B and subgroups H of G. This extends the Galois correspondence for reduced crossed products by discrete group actions established by Cameron-Smith. We relate it to the Galois correspondences of Izumi and Mukohara for fixed-point algebras of actions of compact abelian groups, and the mixed inclusion of a fixed-point algebra in a reduced crossed product considered by Echterhoff-Rørdam. In addition, using a recent result of Geffen-Ursu, we show that the inclusion of a fixed-point subalgebra A⊆ B of an action of 𝕋 or ℤ/p for a square-free number p>0 is aperiodic if and only if A detects ideals in B. We apply this to give examples of C*-irreducible inclusions coming from Cuntz-Pimsner algebras, including crossed products by endomorphisms or transfer operators. In particular, we characterise when a core subalgebra of a graph C*-algebra is C*-irreducible. Lastly, we show that a general regular topologically graded C*-inclusion A⊆ B is aperiodic and has a unique pseudo-expectation provided A detects ideals in all intermediate C*-algebras of B. This partially answers a question by Pitts-Zarikian.
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BibTeX
@misc{kwasniewski2026cirreducible,
title = {{C*-irreducible regular inclusions, Galois correspondence and aperiodicity}},
author = {B. K. Kwaśniewski and R. Meyer},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27518},
eprint = {2609.27518},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27518},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - C*-irreducible regular inclusions, Galois correspondence and aperiodicity
AU - B. K. Kwaśniewski
AU - R. Meyer
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.27518
UR - https://arxiv.org/abs/2609.27518
AB - We characterise C*-irreducible regular C*-inclusions using a number of different conditions considered by different authors. In particular, we show that all C*-irreducible regular inclusions A⊆ B are modelled by outer Fell bundles (B_g)_g∈ G over discrete groups with a simple unit fibre A=B_1. In this case, we prove a bijection between intermediate C*-algebras A⊆ C ⊆ B and subgroups H of G. This extends the Galois correspondence for reduced crossed products by discrete group actions established by Cameron-Smith. We relate it to the Galois correspondences of Izumi and Mukohara for fixed-point algebras of actions of compact abelian groups, and the mixed inclusion of a fixed-point algebra in a reduced crossed product considered by Echterhoff-Rørdam. In addition, using a recent result of Geffen-Ursu, we show that the inclusion of a fixed-point subalgebra A⊆ B of an action of 𝕋 or ℤ/p for a square-free number p>0 is aperiodic if and only if A detects ideals in B. We apply this to give examples of C*-irreducible inclusions coming from Cuntz-Pimsner algebras, including crossed products by endomorphisms or transfer operators. In particular, we characterise when a core subalgebra of a graph C*-algebra is C*-irreducible. Lastly, we show that a general regular topologically graded C*-inclusion A⊆ B is aperiodic and has a unique pseudo-expectation provided A detects ideals in all intermediate C*-algebras of B. This partially answers a question by Pitts-Zarikian.
ER -