Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.04548

Cartan inverse semigroups and twisted Steinberg algebras

Nathan Brownlowe·Lisa Orloff Clark·Ying-Fen Lin·Lynnel D. Naingue
View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

math.OAmath.RA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

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

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Aperiodic automorphisms of the Razak--Jacelon algebra

We completely classify aperiodic automorphisms of the Razak--Jacelon algebra $\mathcal{W}$ up to outer conjugacy. As a technical ingredient, based on the works of Connes, Herman--Ocneanu, Kishimoto, and Sato, we introduce a notion of weak stability for automorphisms of simple separable monotracial C$^*$-algebras. We also consider automorphisms of UHF algebras of infinite type and the Jiang--Su algebra. In particular, we classify aperiodic ``UHF-absorbing'' automorphisms of UHF algebras of infinite type up to outer conjugacy.

math.OA↗
arXiv2026

On the Isomorphism of Semicircular Algebras

Building on OpenAI's proof of isomorphism of free group factors, we give a self-contained proof of a corresponding result for $C^{*}$-algebras generated by free semicircular systems: the isomorphism class of a $C^{*}$-algebra generated by a semicircular $n$-tuple does not depend on $n\in\{2,3,\dots,+\infty\}$. Similarly, reduced free products of $n$ copies of the Jiang--Su algebra are isomorphic for different $n$ in this range. While reduced free group $C^{*}$-algebras of different ranks are not isomorphic (by a result of Pimsner--Voiculescu), we show that they have strong free absorption properties. At the heart of the proof is an automorphism approximation property for inclusions of $C^{*}$- and $W^{*}$-algebras; a relative version of that property was previously used to derive free complementation results.

math.OA↗
arXiv2026

Spectral truncations of SU(2) and the spectral localizer

We study operator systems arising from Peter-Weyl and Dirac spectral truncations of $\mathrm{SU}(2)$ in the sense of Connes-Van Suijlekom. We determine operator system invariants for these systems, including the $C^*$-envelope and uniform bounds on the propagation number. For Peter-Weyl truncations, we find $\mathrm{prop} \leq 2$ for all connected compact Lie groups, generalizing known results for the torus. We also carry out an analysis of the spectral localizer by Loring and Schulz-Baldes on $\mathrm{SU}(2)$. In this analysis, we find numerical evidence that an index map for operator systems previously proposed by Van Suijlekom only captures the right information in a suitable interval of the parameter $κ$, instead of the limit $κ\downarrow 0$.

math.OA↗