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Cartan inverse semigroups and twisted Steinberg algebras

  • Nathan Brownlowe
  • Lisa Orloff Clark
  • Ying-Fen Lin
  • Lynnel D. Naingue
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Abstract

We introduce the notion of a Cartan inverse semigroup and an (algebraic) semi-Cartan subalgebra in an R-algebra A, providing a purely algebraic analogue of Cartan semigroups and semi-Cartan subalgebras in a C*-algebra. This extends the theory of quasi-Cartan subalgebras, shifting the focus from designated subalgebras to designated subsemigroups. Our main result characterises twisted Steinberg algebras in terms of the existence of a Cartan inverse semigroup: we show that an R-algebra admits a Cartan inverse semigroup if and only if it is isomorphic to a twisted Steinberg algebra. Moreover, when our construction is applied to a twisted Steinberg algebra over an ample Hausdorff groupoid, the constructed twist is naturally isomorphic to the original one.

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Cite

BibTeXRIS
BibTeX
@misc{brownlowe2026cartan,
  title = {{Cartan inverse semigroups and twisted Steinberg algebras}},
  author = {Nathan Brownlowe and Lisa Orloff Clark and Ying-Fen Lin and Lynnel D. Naingue},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.04548},
  eprint = {2610.04548},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.04548},
}
RIS
TY  - UNPB
TI  - Cartan inverse semigroups and twisted Steinberg algebras
AU  - Nathan Brownlowe
AU  - Lisa Orloff Clark
AU  - Ying-Fen Lin
AU  - Lynnel D. Naingue
PY  - 2026
DA  - 2026/10/03
PB  - arXiv
DO  - 10.48550/arxiv.2610.04548
UR  - https://arxiv.org/abs/2610.04548
AB  - We introduce the notion of a Cartan inverse semigroup and an (algebraic) semi-Cartan subalgebra in an R-algebra A, providing a purely algebraic analogue of Cartan semigroups and semi-Cartan subalgebras in a C*-algebra. This extends the theory of quasi-Cartan subalgebras, shifting the focus from designated subalgebras to designated subsemigroups. Our main result characterises twisted Steinberg algebras in terms of the existence of a Cartan inverse semigroup: we show that an R-algebra admits a Cartan inverse semigroup if and only if it is isomorphic to a twisted Steinberg algebra. Moreover, when our construction is applied to a twisted Steinberg algebra over an ample Hausdorff groupoid, the constructed twist is naturally isomorphic to the original one.
ER  -