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John Quigg

Publications and source records attributed to John Quigg.

At least 19 recordsLinked to original sources
arXiv2026

Morita equivalence of graded C*-algebras

We define two notions of Morita equivalence for graded C*-algebras (graded Morita equivalence and homogeneous Morita equivalence) and provide Brown-Green-Rieffel Stabilization Type Theorems for both notions of graded equivalence. We apply our results to finite regular graphs by establishing an explicit connection between graded C*-algebras and coactions. Lastly, we incorporate Cartan subalgebras with totally disconnected spectra and obtain Brown-Green-Rieffel Stabilization Type Theorems for these cases.

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arXiv2026

$k$-graph algebras are iterated Cuntz-Pimsner algebras -- from the bottom up

We introduce a new method of expressing a $k$-graph $C^*$-algebra as a Cuntz-Pimsner algebra. Kumjian, Pask, and Sims have done this directly, using a linking algebra approach and a $(k-1)$-graph algebra. This can be iterated downward. Our process, on the other hand, starts at the bottom, with Pimsner's theorem for graph algebras, and iterates upward. We actually work with product systems over $\mathbb N^k$, and the result for $k$-graphs is a special case. Our iteration step involves a ``decategorization'' of a recent theorem showing that the Cuntz-Pimsner construction is functorial at the level of ``enchilada categories''.

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arXiv2026

Comparing Two Notions of Coaction Invariance of Ideals in $\mathrm{C}^*$-Algebras

Given a coaction $\delta$ of a locally compact group $G$ on a $\mathrm{C}^*$-algebra $A$, we study the relationship between two different forms of coaction invariance of ideals of $A$ and the ideals of the corresponding crossed product $\mathrm{C}^*$-algebra $A \rtimes_{\delta} G$. In particular, we characterize when these two notions of invariance are equivalent.

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arXiv2025

Gluing topological graph C*-algebras

We introduce regular closed subgraphs of Katsura's topological graphs and use them to generalize the notion of an adjunction space from topology. Our construction attaches a topological graph onto another via a regular factor map. We prove that under suitable assumptions the C*-algebra of the adjunction graph is a pullback of the C*-algebras of the topological graphs being glued. Our results generalize certain pushout-to-pullback theorems proved in the context of discrete directed graphs. Our theorem applied to homeomorphism C*-algebras recovers a special case of the well-known result stating that pullbacks of $\mathbb{Z}$-C*-algebras induce pullbacks of the respective crossed product C*-algebras. Furthermore, we show that the C*-algebras of odd-dimensional quantum balls of Hong and Szyma\'nski (which are known not to be graph C*-algebras) are topological graph C*-algebras and we recover the pullback structure of C*-algebras of odd-dimensional quantum spheres by gluing the topological graphs associated to the C*-algebras of the corresponding odd-dimensional quantum balls.

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arXiv2024

Fell bundle ladder

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

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arXiv2024

Coactions of compact groups on $M_n$

We prove that every coaction of a compact group on a finite-dimensional $C^*$-algebra is associated with a Fell bundle. Every coaction of a compact group on a matrix algebra is implemented by a unitary operator. A coaction of a compact group on $M_n$ is inner if and only if its fixed-point algebra has an abelian $C^*$-subalgebra of dimension $n$. Investigating the existence of effective ergodic coactions on $M_n$ reveals that $\operatorname{SO}(3)$ has them, while $\operatorname{SU}(2)$ does not. We give explicit examples of the two smallest finite nonabelian groups having effective ergodic coactions on $M_n$.

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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

Strong Pedersen rigidity for coactions of compact groups

We prove a version of Pedersen's outer conjugacy theorem for coactions of compact groups, which characterizes outer conjugate coactions of a compact group in terms of properties of the dual actions. In fact, we show that every isomorphism of a dual action comes from a unique outer conjugacy of a coaction, which in this context should be called strong Pedersen rigidity. We promote this to a category equivalence.

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arXiv2022

Coactions on C*-algebras and universal properties

It is well-known that the maximalization of a coaction of a locally compact group on a C*-algebra enjoys a universal property. We show how this important property can be deduced from a categorical framework by exploiting certain properties of the maximalization functor for coactions. We also provide a dual proof for the universal property of normalization of coactions.

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arXiv2022

Gauge-invariant uniqueness theorems for $P$-graphs

We prove a version of the result in the title that makes use of maximal coactions in the context of discrete groups. Earlier Gauge-Invariant Uniqueness theorems for $C^*$-algebras associated to $P$-graphs and similar $C^*$-algebras exploited a property of coactions known as normality. In the present paper, the view point is that maximal coactions provide a more natural starting point to state and prove such uniqueness theorems. A byproduct of our approach consists of an abstract characterization of co-universal representations for a Fell bundle over a discrete group.

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arXiv2021

The Modular Stone-von Neumann Theorem

In this paper, we use the tools of nonabelian duality to formulate and prove a far-reaching generalization of the Stone-von Neumann Theorem to modular representations of actions and coactions of locally compact groups on elementary $ C^{\ast} $-algebras. This greatly extends the Covariant Stone-von Neumann Theorem for Actions of Abelian Groups recently proven by L. Ismert and the second author. Our approach is based on a new result about Hilbert $ C^{\ast} $-modules that is simple to state yet is widely applicable and can be used to streamline many previous arguments, so it represents an improvement -- in terms of both efficiency and generality -- in a long line of results in this area of mathematical physics that goes back to J. von Neumann's proof of the classical Stone-von Neumann Theorem.

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arXiv2021

R-coactions on $C^*$-algebras

We give the beginnings of the development of a theory of what we call "R-coactions" of a locally compact group on a $C^*$-algebra. These are the coactions taking values in the maximal tensor product, as originally proposed by Raeburn. We show that the theory has some gaps as compared to the more familiar theory of standard coactions. However, we indicate how we needed to develop some of the basic properties of R-coactions as a tool in our program involving the use of coaction functors in the study of the Baum-Connes conjecture.

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arXiv2021

Tensor $D$ coaction functors

We develop an approach, using what we call "tensor $D$ coaction functors", to the "$C$-crossed-product" functors of Baum, Guentner, and Willett. We prove that the tensor $D$ functors are exact, and identify the minimal such functor. This continues our program of applying coaction functors as a tool in the Baum-Guentner-Willett-Buss-Echterhoff campaign to attempt to "fix" the Baum-Connes conjecture.

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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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arXiv2019

Tensor-product coaction functors

For a discrete group $G$, we develop a `$G$-balanced tensor product' of two coactions $(A,δ)$ and $(B,ε)$, which takes place on a certain subalgebra of the maximal tensor product $A\otimes_{\max} B$. Our motivation for this is that we are able to prove that given two actions of $G$, the dual coaction on the crossed product of the maximal-tensor-product action is isomorphic to the $G$-balanced tensor product of the dual coactions. In turn, our motivation for this is to give an analogue, for coaction functors, of a crossed-product functor originated by Baum, Guentner, and Willett, and further developed by Buss, Echterhoff, and Willett, that involves tensoring an action with a fixed action $(C,γ)$, then forming the image inside the crossed product of the maximal-tensor-product action. We prove that composing our tensor-product coaction functor with the full crossed product of an action reproduces the tensor-crossed-product functor of Baum, Guentner, and Willett. We prove that every such tensor-product coaction functor is exact, thereby recovering the analogous result for the tensor-crossed-product functors of Baum, Guentner, and Willett. When $(C,γ)$ is the action by translation on $\ell^\infty(G)$, we prove that the associated tensor-product coaction functor is minimal, generalizing the analogous result of Buss, Echterhoff, and Willett for tensor-crossed-product functors.

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