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arXiv · 2610.04548

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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Nathan Brownlowe, Lisa Orloff Clark, Ying-Fen Lin, Lynnel D. Naingue. 2026-10-03. Cartan inverse semigroups and twisted Steinberg algebras. https://arxiv.org/abs/2610.04548

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