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Dana P. Williams

Publications and source records attributed to Dana P. Williams.

At least 19 recordsLinked to original sources
arXiv2026

Nuclear dimension of groupoid C*-algebras with large abelian isotropy, with applications to C*-algebras of directed graphs and twists

We characterise when the C*-algebra C*(G) of a locally compact and Hausdorff groupoid G is subhomogeneous, that is, when its irreducible representations have bounded finite dimension; if so we establish a bound for its nuclear dimension in terms of the topological dimensions of the unit space of the groupoid and the spectra of the primitive ideal spaces of the isotropy subgroups. For an etale groupoid G, we also establish a bound on the nuclear dimension of its C*-algebra provided the quotient of G by its isotropy subgroupid has finite dynamic asymptotic dimension in the sense of Guentner, Willet and Yu. Our results generalise those of C. Böncicke and K. Li to groupoids with large isotropy, including graph groupoids of directed graphs. We find that all graph C*-algebras that are stably finite have nuclear dimension at most 1. We also show that the nuclear dimension of the C*-algebra of a twist over G has the same bound on the nuclear dimension as for C*(G) and the twisted groupoid C*-algebra.

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arXiv2025

Fell bundle ladder

We use the Ladder Technique to establish bijections between the ideals of related Fell bundles.

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arXiv2025

Non-traditional Cartan subalgebras in twisted groupoid C*-algebras

Well-known work of Renault shows that if $\mathcal{E}$ is a twist over a second countable, effective, étale groupoid $G$, then there is a naturally associated Cartan subalgebra of the reduced twisted groupoid C*-algebra $C^*_{r}(G; E)$, and that every Cartan subalgebra of a separable C*-algebra arises in this way. However twisted C*-algebras of non-effective groupoids $G$ can also possess Cartan subalgebras: In work by the first author together with Gillaspy, Norton, Reznikoff, and Wright, sufficient conditions on a subgroupoid $S$ of $G$ were found that ensure that $S$ gives rise to a Cartan subalgebra in the cocycle-twisted C*-algebra of $G$. In this paper, we extend these results to general twists $\mathcal{E}$, and we refine the conditions on the subgroupoid for $C^*_{r}(S;\mathcal{E}_S)$ to be a Cartan subalgebra of $C^*_{r}(G;\mathcal{E})$.

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arXiv2024

Bijections Between Sets of Invariant Ideals, Via the Ladder Technique

We present a new method of establishing a bijective correspondence - in fact, a lattice isomorphism - between action- and coaction-invariant ideals of C*-algebras and their crossed products by a fixed locally compact group. It is known that such a correspondence exists whenever the group is amenable; our results hold for any locally compact group under a natural form of coaction invariance.

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arXiv2022

Renault's $j$-map for Fell bundle $C^*$-algebras

If $p \colon \mathcal B\to G$ is a Fell bundle over an étale groupoid, then we show that there is an norm reducing injective linear map $j \colon C^*_r(G;\mathcal B)\to Γ_{0}(G;\mathcal B)$ generalizing the well know map $j \colon C^*_{r}(G)\to C_{0}(G)$ in the case of an étale groupoid.

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arXiv2021

Groupoid Semidirect Product Fell Bundles II- Principal Actions and Stabilization

Given a free and proper action of a groupoid on a Fell bundle (over another groupoid), we give an equivalence between the semidirect-product and the generalized-fixed-point Fell bundles, generalizing an earlier result where the action was by a group. As an application, we show that the Stabilization Theorem for Fell bundles over groupoids is essentially another form of crossed-product duality.

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arXiv2021

The Primitive Ideal Space of Groupoid C*-Algebras for Groupoids with Abelian Isotropy

We study the topology of the primitive ideal space of groupoid C*-algebras for groupoids with abelian isotropy. Our results include the known results for action groupoids with abelian stabilizers. Furthermore, we obtain complete results when the isotropy map is continuous except for jump discontinuities, and also when $G$ is a unit space fixing extension of a proper groupoid by an abelian group bundle. We hope that our methods will be a springboard to further results of this type.

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arXiv2021

Pushouts of extensions of groupoids by bundles of abelian groups

We analyse extensions $Σ$ of groupoids $G$ by bundles $A$ of abelian groups. We describe a pushout construction for such extensions, and use it to describe the extension group of a given groupoid $G$ by a given bundle $A$. There is a natural action of $Σ$ on the dual of $A$, yielding a corresponding transformation groupoid. The pushout of this transformation groupoid by the natural map from the fibre product of $A$ with its dual to the Cartesian product of the dual with the circle is a twist over the transformation groupoid resulting from the action of $G$ on the dual of $A$. We prove that the full $C^*$-algebra of this twist is isomorphic to the full $C^*$-algebra of $Σ$, and that this isomorphism descends to an isomorphism of reduced algebras. We give a number of examples and applications.

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arXiv2020

C*-Algebras of extensions of groupoids by group bundles

Given a normal subgroup bundle $\mathcal A$ of the isotropy bundle of a groupoid $Σ$, we obtain a twisted action of the quotient groupoid $Σ/\mathcal A$ on the bundle of group $C^*$-algebras determined by $\mathcal A$ whose twisted crossed product recovers the groupoid $C^*$-algebra $C^*(Σ)$. Restricting to the case where $\mathcal A$ is abelian, we describe $C^*(Σ)$ as the $C^*$-algebra associated to a $\mathbf T$-groupoid over the tranformation groupoid obtained from the canonical action of $Σ/\mathcal A$ on the Pontryagin dual space of $\mathcal A$. We give some illustrative examples of this result.

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arXiv2018

The Dixmier-Douady Classes of Certain Groupoid $C^*$-Algebras with Continuous Trace

Given a locally compact abelian group $G$, we give an explicit formula for the Dixmier--Douady invariant of the $C^*$-algebra of the groupoid extension associated to a Čech $2$-cocycle in the sheaf of germs of continuous $G$-valued functions. We then exploit the blow-up construction for groupoids to extend this to some more general central extensions of étale equivalence relations.

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arXiv2016

A Stabilization Theorem for Fell Bundles over groupoids

We study the $C^*$-algebras associated to upper-semicontinuous Fell bundles over second-countable Hausdorff groupoids. Based on ideas going back to the Packer--Raeburn "Stabilization Trick," we construct from each such bundle a groupoid dynamical system whose associated Fell bundle is equivalent to the original bundle. The upshot is that the full and reduced $C^*$-algebras of any saturated upper-semicontinuous Fell bundle are stably isomorphic to the full and reduced crossed products of an associated dynamical system. We apply our results to describe the lattice of ideals of the $C^*$-algebra of a continuous Fell-bundle by applying Renault's results about the ideals of the $C^*$-algebras of groupoid crossed products. In particular, we discuss simplicity of the Fell-bundle $C^*$-algebra of a bundle over $G$ in terms of an action, described by the first and last named authors, of $G$ on the primitive-ideal space of the $C^*$-algebra of the part of the bundle sitting over the unit space. We finish with some applications to twisted $k$-graph algebras, where the components of our results become more concrete.

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arXiv2016

Cartan subalgebras in C*-algebras of Hausdorff etale groupoids

The reduced $C^*$-algebra of the interior of the isotropy in any Hausdorff étale groupoid $G$ embeds as a $C^*$-subalgebra $M$ of the reduced $C^*$-algebra of $G$. We prove that the set of pure states of $M$ with unique extension is dense, and deduce that any representation of the reduced $C^*$-algebra of $G$ that is injective on $M$ is faithful. We prove that there is a conditional expectation from the reduced $C^*$-algebra of $G$ onto $M$ if and only if the interior of the isotropy in $G$ is closed. Using this, we prove that when the interior of the isotropy is abelian and closed, $M$ is a Cartan subalgebra. We prove that for a large class of groupoids $G$ with abelian isotropy---including all Deaconu--Renault groupoids associated to discrete abelian groups---$M$ is a maximal abelian subalgebra. In the specific case of $k$-graph groupoids, we deduce that $M$ is always maximal abelian, but show by example that it is not always Cartan.

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arXiv2015

Amenability of Groupoids Arising from Partial Semigroup Actions and Topological Higher Rank Graphs

We consider the amenability of groupoids $G$ equipped with a group valued cocycle $c:G\to Q$ with amenable kernel $c^{-1}(e)$. We prove a general result which implies, in particular, that $G$ is amenable whenever $Q$ is amenable and if there is countable set $D\subset G$ such that $c(G^{u})D=Q$ for all $u\in G^{(0)}$. We show that our result is applicable to groupoids arising from partial semigroup actions. We explore these actions in detail and show that these groupoids include those arising from directed graphs, higher rank graphs and even topological higher rank graphs. We believe our methods yield a nice alternative groupoid approach to these important constructions.

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arXiv2015

The primitive ideals of some étale groupoid C*-algebras

Consider the Deaconu-Renault groupoid of an action of a finitely generated free abelian monoid by local homeomorphisms of a locally compact Hausdorff space. We catalogue the primitive ideals of the associated groupoid C*-algebra. For a special class of actions we describe the Jacobson topology.

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