arXiv subjects

Lisa Orloff Clark

Publications and source records attributed to Lisa Orloff Clark.

At least 19 recordsLinked to original sources

arXiv2026

Groupoids of finitely aligned higher-rank graphs via filters and graph morphisms

Higher-rank graphs are certain small categories that are a higher-dimensional generalisation of both directed graphs and free monoids. Path and boundary-path groupoids of finitely aligned higher-rank graphs are often constructed using either filters or graph morphisms. We generalise the graph morphism approach to finitely aligned P-graphs where (Q, P) is a weakly quasi-lattice ordered group, and we show the filter approach and the graph morphism approach yield isomorphic path and boundary-path groupoids. To do this, we define conjugacy of partial monoid actions such that conjugate actions have isomorphic semidirect product groupoids. Combining our results with others in the literature, we survey many isomorphic presentations of path and boundary-path groupoids at different levels of generality.

math.RA↗

arXiv2026

Singular ideals over arbitrary fields for the cyclic-headed snakes

We study the Steinberg algebras with coefficients in an arbitrary field K for the cyclic-headed snake groupoids, which are basic examples of non-Hausdorff groupoids. We are particularly interested in elements of this algebra that are no longer continuous, known as singular functions. These functions form an ideal, which may contain proper subsets that are themselves ideals of the Steinberg algebra. We provide three conditions under which such 'proper subset' ideals exist: first, when the number of heads of the snake divides the characteristic of the base field; second, when the base field is of non-prime characteristic; and third, when certain cyclotomic polynomials split over the base field. We also show the existence of many further subset ideals not covered by these conditions. We fully explore the cases of the two- and three-headed snakes. In the three-headed snake, we prove that the ideal of singular functions properly contains non-zero ideals of the Steinberg algebra if, and only if, the base field K is a splitting field of x^2 + x + 1, the third cyclotomic polynomial. Consequently, there are always proper subset ideals when K has characteristic a prime not congruent to -1 mod 3.

math.RA↗

arXiv2026

A dichotomy for inverse-semigroup crossed products via dynamical Cuntz semigroups

We characterise stable finiteness and pure infiniteness of the essential crossed product of a C*-algebra by an action of an inverse semigroup. Under additional assumptions, we prove a stably finite / purely infinite dichotomy. Our main technique is the development, using an induced action, of a ``dynamical Cuntz semigroup'' that is a subquotient of the usual Cuntz semigroup. We prove that the essential crossed product is stably finite / purely infinite if and only if the dynamical Cuntz semigroup admits / does not admit a nontrivial state. Indeed, a retract of our dynamical Cuntz semigroup suffices to prove the dichotomy. Our results generalise those by Rainone on crossed products of groups acting by automorphisms of a C*-algebra, and we recover results by Kwaśniewski--Meyer--Prasad on C*-algebras of non-Hausdorff groupoids.

math.OA↗

arXiv2026

Topologically free non-Hausdorff groupoids

We study three conditions that control the behaviour of isotropy in étale groupoids, and their relationships under the additional assumptions of second-countability and Hausdorffness. We examine a number of examples that show these properties are distinct. Working under the assumption of the Zermelo-Fraenkel axioms, excluding choice, we then examine an alternate characterization of topological freeness, first introduced by Anantharaman-Delaroche, in the non-Hausdorff setting. Finally, we prove an equivalence between the Baire Category Theorem and an étale groupoid theorem, along with similar equivalences to other weakenings of the Axiom of Choice.

math.OA↗

arXiv2025

Generalised Twisted Groupoids and their C*-algebras

We consider a locally compact Hausdorff groupoid $G$, and twist by a more general locally compact Hausdorff abelian group $Γ$ rather than the complex unit circle $\mathbb{T}$. We investigate the construction of $C^*$-algebras in analogue to the usual twisted groupoid $C^*$-algebras, and we show that, in fact, any $Γ$-twisted groupoid $C^*$-algebra is isomorphic to a usual twisted groupoid $C^*$-algebra.

math.OA↗

arXiv2025

On Graded Quasi-Cartan Pairs and Twisted Steinberg Algebras

We generalise recent results about quasi-Cartan, Cartan and diagonal subalgebras by introducing graded versions. We show that there is a correspondence between graded algebraic quasi-Cartan/ Cartan/ diagonal pairs and certain graded twisted Steinberg algebras and that the associated graded discrete twist is unique. Our results include all discrete group algebras, and so are more general than the ungraded version.

math.RA↗

arXiv2025

Cartan semigroups and twisted groupoid C*-algebras

We prove that twisted groupoid C*-algebras are characterised, up to isomorphism, by having Cartan semigroups, a natural generalisation of normaliser semigroups of Cartan subalgebras. This extends the classic Kumjian-Renault theory to general twisted étale groupoid C*-algebras, even non-reduced C*-algebras of non-effective groupoids.

math.OA↗

arXiv2025

On the socle of a class of Steinberg algebras

We study minimal left ideals in Steinberg algebras of Hausdorff groupoids. We establish a relationship between minimal left ideals in the algebra and open singletons in the unit space of the groupoid. We apply this to obtain results about the socle of Steinberg algebras under certain hypotheses. This encompasses known results about Leavitt path algebras and improves on Kumjian-Pask algebra results to include higher-rank graphs that are not row-finite.

math.OA↗

arXiv2024

Intermediate Subalgebras of Cartan embeddings in rings and C*-algebras

Let $D \subseteq A$ be a quasi-Cartan pair of algebras. Then there exists a unique discrete groupoid twist $Σ\to G$ whose twisted Steinberg algebra is isomorphic to $A$ in a way that preserves $D$. In this paper, we show there is a lattice isomorphism between wide open subgroupoids of $G$ and subalgebras $C$ such that $D\subseteq C\subseteq A$ and $D \subseteq C$ is a quasi-Cartan pair. We also characterise which algebraic diagonal/algebraic Cartan/quasi-Cartan pairs have the property that every subalgebra $C$ with $D\subseteq C\subseteq A$ has $D \subseteq C$ a diagonal/Cartan/quasi-Cartan pair. In the diagonal case, when the coefficient ring is a field, it is all of them. Beyond that, only pairs that are close to being diagonal have this property. We then apply our techniques to C*-algebraic inclusions and give a complete characterization of which Cartan pairs $D \subseteq A$ have the property that every C*-subalgebra $C$ with $D\subseteq C\subseteq A$ has $D \subseteq C$ a Cartan pair.

math.RA↗

arXiv2024

Representing topological full groups in Steinberg algebras and C*-algebras

We study the natural representation of the topological full group of an ample Hausdorff groupoid in the groupoid's complex Steinberg algebra and in its full and reduced C*-algebras. We characterise precisely when this representation is injective and show that it is rarely surjective. We then restrict our attention to discrete groupoids, which provide unexpected insight into the behaviour of the representation of the topological full group in the full and reduced groupoid C*-algebras. We show that the image of the representation is not dense in the full groupoid C*-algebra unless the groupoid is a group, and we provide an example showing that the image of the representation may still be dense in the reduced groupoid C*-algebra even when the groupoid is not a group.

math.OA↗

arXiv2023

The local bisection hypothesis for twisted groupoid C*-algebras

In this note, we present criteria that are equivalent to a locally compact Hausdorff groupoid $G$ being effective. One of these conditions is that $G$ satisfies the "C*-algebraic local bisection hypothesis"; that is, that every normaliser in the reduced twisted groupoid C*-algebra is supported on an open bisection. The semigroup of normalisers plays a fundamental role in our proof, as does the semigroup of normalisers in cyclic group C*-algebras.

math.OA↗

arXiv2023

Equivalence of definitions of AF groupoid

We prove the equivalence of two definitions of AF groupoid in the literature: one by Renault and the other by Farsi, Kumjian, Pask and Sims. In both definitions, an AF groupoid is an increasing union of more basic groupoids, called elementary groupoids. Surprisingly, the two definitions of elementary groupoid are not equivalent; they coincide if and only if the local homeomorphism that characterises them is a covering map.

math.OA↗

arXiv2023

Inclusions of C*-algebras of graded groupoids

We consider a locally compact Hausdorff groupoid $G$ which is graded over a discrete group. Then the fibre over the identity is an open and closed subgroupoid $G_e$. We show that both the full and reduced C*-algebras of this subgroupoid embed isometrically into the full and reduced C*-algebras of $G$; this extends a theorem of Kaliszewski--Quigg--Raeburn from the étale to the non-étale setting. As an application we show that the full and reduced C*-algebras of $G$ are topologically graded in the sense of Exel, and we discuss the full and reduced C*-algebras of the associated bundles.

math.OA↗

arXiv2022

A Steinberg algebra approach to étale groupoid C*-algebras

We construct the full and reduced C*-algebras of an ample groupoid from its complex Steinberg algebra. We also show that our construction gives the same C*-algebras as the standard constructions. In the last section, we consider an arbitrary locally compact, second-countable, étale groupoid, possibly non-Hausdorff. Using the techniques developed for Steinberg algebras, we show that every $*$-homomorphism from Connes' space of functions to $B(\mathcal{H})$ is automatically I-norm bounded. Previously, this was only known for Hausdorff groupoids.

math.OA↗

arXiv2021

Reconstruction of twisted Steinberg algebras

We show how to recover a discrete twist over an ample Hausdorff groupoid from a pair consisting of an algebra and what we call a quasi-Cartan subalgebra. We identify precisely which twists arise in this way (namely, those that satisfy the local bisection hypothesis), and we prove that the assignment of twisted Steinberg algebras to such twists and our construction of a twist from a quasi-Cartan pair are mutually inverse. We identify the algebraic pairs that correspond to effective groupoids and to principal groupoids. We also indicate the scope of our results by identifying large classes of twists for which the local bisection hypothesis holds automatically.

math.RA↗

arXiv2021

Filtering germs: Groupoids associated to inverse semigroups

We investigate various groupoids associated to an arbitrary inverse semigroup with zero. We show that the groupoid of filters with respect to the natural partial order is isomorphic to the groupoid of germs arising from the standard action of the inverse semigroup on the space of idempotent filters. We also investigate the restriction of this isomorphism to the groupoid of tight filters and to the groupoid of ultrafilters.

math.RA↗

arXiv2021

Twisted Steinberg algebras

We introduce twisted Steinberg algebras over a commutative unital ring $R$. These generalise Steinberg algebras and are a purely algebraic analogue of Renault's twisted groupoid C*-algebras. In particular, for each ample Hausdorff groupoid $G$ and each locally constant $2$-cocycle $σ$ on $G$ taking values in the units $R^\times$, we study the algebra $A_R(G,σ)$ consisting of locally constant compactly supported $R$-valued functions on $G$, with convolution and involution "twisted" by $σ$. We also introduce a "discretised" analogue of a twist $Σ$ over a Hausdorff étale groupoid $G$, and we show that there is a one-to-one correspondence between locally constant $2$-cocycles on $G$ and discrete twists over $G$ admitting a continuous global section. Given a discrete twist $Σ$ arising from a locally constant $2$-cocycle $σ$ on an ample Hausdorff groupoid $G$, we construct an associated twisted Steinberg algebra $A_R(G;Σ)$, and we show that it coincides with $A_R(G,σ^{-1})$. Given any discrete field $\mathbb{F}_d$, we prove a graded uniqueness theorem for $A_{\mathbb{F}_d}(G,σ)$, and under the additional hypothesis that $G$ is effective, we prove a Cuntz--Krieger uniqueness theorem and show that simplicity of $A_{\mathbb{F}_d}(G,σ)$ is equivalent to minimality of $G$.

math.RA↗

arXiv2020

Reconstructing Etale Groupoids from Semigroups

We unify various étale groupoid reconstruction theorems such as: 1) Kumjian-Renault's reconstruction from a groupoid C*-algebra. 2) Exel's reconstruction from an ample inverse semigroup. 3) Steinberg's reconstruction from a groupoid ring. 4) Choi-Gardella-Thiel's reconstruction from a groupoid L^p-algebra. We do this by working with certain bumpy semigroups S of functions defined on an étale groupoid G. The semigroup structure of S together with the diagonal subsemigroup D then yields a natural domination relation < on S. The groupoid of <-ultrafilters is then isomorphic to the original groupoid G.

math.OA↗