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Becky Armstrong

Publications and source records attributed to Becky Armstrong.

12 recordsLinked to original sources
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 Kwa\'sniewski--Meyer--Prasad on C*-algebras of non-Hausdorff groupoids.

math.OA↗
arXiv2023

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↗
arXiv2022

A uniqueness theorem for twisted groupoid C*-algebras

We present a uniqueness theorem for the reduced C*-algebra of a twist $\mathcal{E}$ over a Hausdorff étale groupoid $\mathcal{G}$. We show that the interior $\mathcal{I}^\mathcal{E}$ of the isotropy of $\mathcal{E}$ is a twist over the interior $\mathcal{I}^\mathcal{G}$ of the isotropy of $\mathcal{G}$, and that the reduced twisted groupoid C*-algebra $C_r^*(\mathcal{I}^\mathcal{G}; \mathcal{I}^\mathcal{E})$ embeds in $C_r^*(\mathcal{G}; \mathcal{E})$. We also investigate the full and reduced twisted C*-algebras of the isotropy groups of $\mathcal{G}$, and we provide a sufficient condition under which states of (not necessarily unital) C*-algebras have unique state extensions. We use these results to prove our uniqueness theorem, which states that a C*-homomorphism of $C_r^*(\mathcal{G}; \mathcal{E})$ is injective if and only if its restriction to $C_r^*(\mathcal{I}^\mathcal{G}; \mathcal{I}^\mathcal{E})$ is injective. We also show that if $\mathcal{G}$ is effective, then $C_r^*(\mathcal{G}; \mathcal{E})$ is simple if and only if $\mathcal{G}$ is minimal.

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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

Simplicity of twisted C*-algebras of Deaconu--Renault groupoids

We consider Deaconu--Renault groupoids associated to actions of finite-rank free abelian monoids by local homeomorphisms of locally compact Hausdorff spaces. We study simplicity of the twisted C*-algebra of such a groupoid determined by a continuous circle-valued groupoid 2-cocycle. When the groupoid is not minimal, this C*-algebra is never simple, so we focus on minimal groupoids. We describe an action of the quotient of the groupoid by the interior of its isotropy on the spectrum of the twisted C*-algebra of the interior of the isotropy. We prove that the twisted groupoid C*-algebra is simple if and only if this action is minimal. We describe applications to crossed products of topological-graph C*-algebras by quasi-free actions.

math.OA↗
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↗
arXiv2021

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 \'etale to the non-\'etale 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↗
arXiv2021

Conjugacy of local homeomorphisms via groupoids and C*-algebras

We investigate dynamical systems consisting of a locally compact Hausdorff space equipped with a partially defined local homeomorphism. Important examples of such systems include self-covering maps, one-sided shifts of finite type and, more generally, the boundary-path spaces of directed and topological graphs. We characterise topological conjugacy of these systems in terms of isomorphisms of their associated groupoids and C*-algebras. This significantly generalises recent work of Matsumoto and of the second- and third-named authors.

math.OA↗
arXiv2018

Product-system models for twisted $C^*$-algebras of topological higher-rank graphs

We use product systems of $C^*$-correspondences to introduce twisted $C^*$-algebras of topological higher-rank graphs. We define the notion of a continuous $\mathbb{T}$-valued $2$-cocycle on a topological higher-rank graph, and present examples of such cocycles on large classes of topological higher-rank graphs. To every proper, source-free topological higher-rank graph $Λ$, and continuous $\mathbb{T}$-valued $2$-cocycle $c$ on $Λ$, we associate a product system $X$ of $C_0(Λ^0)$-correspondences built from finite paths in $Λ$. We define the twisted Cuntz--Krieger algebra $C^*(Λ,c)$ to be the Cuntz--Pimsner algebra $\mathcal{O}(X)$, and we define the twisted Toeplitz algebra $\mathcal{T} C^*(Λ,c)$ to be the Nica--Toeplitz algebra $\mathcal{NT}(X)$. We also associate to $Λ$ and $c$ a product system $Y$ of $C_0(Λ^\infty)$-correspondences built from infinite paths. We prove that there is an embedding of $\mathcal{T} C^*(Λ,c)$ into $\mathcal{NT}(Y)$, and an isomorphism between $C^*(Λ,c)$ and $\mathcal{O}(Y)$.

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