arXiv subjects

John Quigg

Publications and source records attributed to John Quigg.

At least 37 records · Page 2Linked to original sources

arXiv2021

Skew products of finitely aligned left cancellative small categories and Cuntz-Krieger algebras

Given a group cocycle on a finitely aligned left cancellative small category (LCSC) we investigate the associated skew product category and its Cuntz-Krieger algebra, which we describe as the crossed product of the Cuntz-Krieger algebra of the original category by an induced coaction of the group. We use our results to study Cuntz-Krieger algebras arising from free actions of groups on finitely aligned LCSC's, and to construct coactions of groups on Exel-Pardo algebras. Finally we discuss the universal group of a small category and connectedness of skew product categories.

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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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arXiv2019

Rigidity theory for $C^*$-dynamical systems and the "Pedersen Rigidity Problem", II

This is a follow-up to a paper with the same title and by the same authors. In that paper, all groups were assumed to be abelian, and we are now aiming to generalize the results to nonabelian groups. The motivating point is Pedersen's theorem, which does hold for an arbitrary locally compact group $G$, saying that two actions $(A,α)$ and $(B,β)$ of $G$ are outer conjugate if and only if the dual coactions $(A\rtimes_αG,\widehatα)$ and $(B\rtimes_βG,\widehatβ)$ of $G$ are conjugate via an isomorphism that maps the image of $A$ onto the image of $B$ (inside the multiplier algebras of the respective crossed products). We do not know of any examples of a pair of non-outer-conjugate actions such that their dual coactions are conjugate, and our interest is therefore exploring the necessity of latter condition involving the images, and we have decided to use the term "Pedersen rigid" for cases where this condition is indeed redundant. There is also a related problem, concerning the possibility of a so-called equivariant coaction having a unique generalized fixed-point algebra, that we call "fixed-point rigidity". In particular, if the dual coaction of an action is fixed-point rigid, then the action itself is Pedersen rigid, and no example of non-fixed-point-rigid coaction is known.

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arXiv2019

On finitely aligned left cancellative small categories, Zappa-Szép products and Exel-Pardo algebras

We consider Toeplitz and Cuntz-Krieger $C^*$-algebras associated with finitely aligned left cancellative small categories. We pay special attention to the case where such a category arises as the Zappa-Szép product of a category and a group linked by a one-cocycle. As our main application, we obtain a new approach to Exel-Pardo algebras in the case of row-finite graphs. We also present some other ways of constructing $C^*$-algebras from left cancellative small categories and discuss their relationship.

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arXiv2018

Subgroup correspondences

For a closed subgroup of a locally compact group the Rieffel induction process gives rise to a $C^*$-correspondence over the $C^*$-algebra of the subgroup. We study the associated Cuntz-Pimsner algebra and show that, by varying the subgroup to be open, compact, or discrete, there are connections with the Exel-Pardo correspondence arising from a cocycle, and also with graph algebras.

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arXiv2017

Rigidity theory for $C^*$-dynamical systems and the "Pedersen Rigidity Problem"

Let $G$ be a locally compact abelian group. By modifying a theorem of Pedersen, it follows that actions of $G$ on $C^*$-algebras $A$ and $B$ are outer conjugate if and only if there is an isomorphism of the crossed products that is equivariant for the dual actions and preserves the images of $A$ and $B$ in the multiplier algebras of the crossed products. The rigidity problem discussed in this paper deals with the necessity of the last condition concerning the images of $A$ and $B$. There is an alternative formulation of the problem: an action of the dual group $\hat G$ together with a suitably equivariant unitary homomorphism of $G$ give rise to a generalized fixed-point algebra via Landstad's theorem, and a problem related to the above is to produce an action of $\hat G$ and two such equivariant unitary homomorphisms of $G$ that give distinct generalized fixed-point algebras. We present several situations where the condition on the images of $A$ and $B$ is redundant, and where having distinct generalized fixed-point algebras is impossible. For example, if $G$ is discrete, this will be the case for all actions of $G$.

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arXiv2017

On Exel-Pardo algebras

We generalize a recent construction of Exel and Pardo, from discrete groups acting on finite directed graphs to locally compact groups acting on topological graphs. To each cocycle for such an action, we construct a $C^*$-correspondence whose associated Cuntz-Pimsner algebra is the analog of the Exel-Pardo $C^*$-algebra.

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arXiv2017

Coaction functors, II

In further study of the application of crossed-product functors to the Baum-Connes Conjecture, Buss, Echterhoff, and Willett introduced various other properties that crossed-product functors may have. Here we introduce and study analogues of these properties for coaction functors, making sure that the properties are preserved when the coaction functors are composed with the full crossed product to make a crossed-product functor. The new properties for coaction functors studied here are functoriality for generalized homomorphisms and the correspondence property. We particularly study the connections with the ideal property. The study of functoriality for generalized homomorphisms requires a detailed development of the Fischer construction of maximalization of coactions with regard to possibly degenerate homomorphisms into multiplier algebras. We verify that all "KLQ" functors arising from large ideals of the Fourier-Stieltjes algebra $B(G)$ have all the properties we study, and at the opposite extreme we give an example of a coaction functor having none of the properties.

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arXiv2016

Ordered invariant ideals of Fourier-Stieltjes algebras

In a recent paper on exotic crossed products, we included a lemma concerning ideals of the Fourier-Stieltjes algebra. Buss, Echterhoff, and Willett have pointed out to us that our proof of this lemma contains an error. In fact, it remains an open question whether the lemma is true as stated. In this note we indicate how to contain the resulting damage. Our investigation of the above question leads us to define two properties \emph{ordered} and \emph{weakly ordered} for invariant ideals of Fourier-Stieltjes algebras, and we initiate a study of these properties.

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arXiv2016

Dualities for maximal coactions

We present a new construction of crossed-product duality for maximal coactions that uses Fischer's work on maximalizations. Given a group $G$ and a coaction $(A,δ)$ we define a generalized fixed-point algebra as a certain subalgebra of $M(A\rtimes_δ G \rtimes_{\widehatδ} G)$, and recover the coaction via this double crossed product. Our goal is to formulate this duality in a category-theoretic context, and one advantage of our construction is that it breaks down into parts that are easy to handle in this regard. We first explain this for the category of nondegenerate *-homomorphisms, and then analogously for the category of $C^*$-correspondences. Also, we outline partial results for the "outer" category, studied previously by the authors.

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arXiv2016

Coaction functors

A certain type of functor on a category of coactions of a locally compact group on C*-algebras is introduced and studied. These functors are intended to help in the study of the crossed-product functors that have been recently introduced in relation to the Baum-Connes conjecture. The most important coaction functors are the ones induced by large ideals of the Fourier-Stieltjes algebra. It is left as an open problem whether the "minimal exact and Morita compatible crossed-product functor" is induced by a large ideal.

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arXiv2016

Exact large ideals of B(G) are downward directed

We prove that if E and F are large ideals of B(G) for which the associated coaction functors are exact, then the same is true for the intersection of E and F. We also give an example of a coaction functor whose restriction to the maximal coactions does not come from any large ideal.

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arXiv2016

Three versions of categorical crossed-product duality

In this partly expository paper we compare three different categories of C*-algebras in which crossed-product duality can be formulated, both for actions and for coactions of locally compact groups. In these categories, the isomorphisms correspond to C*-algebra isomorphisms, imprimitivity bimodules, and outer conjugacies, respectively. In each case, a variation of the fixed-point functor that arises from classical Landstad duality is used to obtain a quasi-inverse for a crossed-product functor. To compare the various cases, we describe in a formal way our view of the fixed-point functor as an "inversion" of the process of forming a crossed product. In some cases, we obtain what we call "good" inversions, while in others we do not. For the outer-conjugacy categories, we generalize a theorem of Pedersen to obtain a fixed-point functor that is quasi-inverse to the reduced-crossed-product functor for actions, and we show that this gives a good inversion. For coactions, we prove a partial version of Pedersen's theorem that allows us to define a fixed-point functor, but the question of whether it is a quasi-inverse for the crossed-product functor remains open.

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arXiv2015

Exotic coactions

If a locally compact group G acts on a C*-algebra B, we have both full and reduced crossed products, and each has a coaction of G. We investigate "exotic" coactions in between, that are determined by certain ideals E of the Fourier-Stieltjes algebra B(G) -- an approach that is inspired by recent work of Brown and Guentner on new C*-group algebra completions. We actually carry out the bulk of our investigation in the general context of coactions on a C*-algebra A. Buss and Echterhoff have shown that not every coaction comes from one of these ideals, but nevertheless the ideals do generate a wide array of exotic coactions. Coactions determined by these ideals E satisfy a certain "E-crossed product duality", intermediate between full and reduced duality. We give partial results concerning exotic coactions, with the ultimate goal being a classification of which coactions are determined by ideals of B(G).

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arXiv2015

Cuntz-Li algebras from a-adic numbers

The a-adic numbers are those groups that arise as Hausdorff completions of noncyclic subgroups of the rational numbers. We give a crossed product construction of (stabilized) Cuntz-Li algebras coming from the a-adic numbers and investigate the structure of the associated algebras. In particular, these algebras are in many cases Kirchberg algebras in the UCT class. Moreover, we prove an a-adic duality theorem, which links a Cuntz-Li algebra with a corresponding dynamical system on the real numbers. The paper also contains an appendix where a nonabelian version of the "subgroup of dual group theorem" is given in the setting of coactions.

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arXiv2015

Ionescu's theorem for higher rank graphs

We will define new constructions similar to the graph systems of correspondences described by Deaconu et al. We will use these to prove a version of Ionescu's theorem for higher rank graphs. Afterwards we will examine the properties of these constructions further and make contact with Yeend's topological k-graphs and the tensor groupoid valued product systems of Fowler and Sims.

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arXiv2015

Destabilization

This partly expository paper first supplies the details of a method of factoring a stable C*-algebra A as B \otimes K in a canonical way. Then it is shown that this method can be put into a categorical framework, much like the crossed-product dualities, and that stabilization gives rise to an equivalence between the nondegenerate category of C*-algebras and a category of "K-algebras". We consider this equivalence as "inverting" the stabilization process, that is, a "destabilization". Furthermore, the method of factoring stable C*-algebras generalizes to Hilbert bimodules, and an analogous category equivalence between the associated enchilada categories is produced, giving a destabilization for C*-correspondences. Finally, we make a connection with (double) crossed-product duality.

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arXiv2015

Properness conditions for actions and coactions

Three properness conditions for actions of locally compact groups on C*-algebras are studied, as well as their dual analogues for coactions. To motivate the properness conditions for actions, the commutative cases (actions on spaces) are surveyed; here the conditions are known: proper, locally proper, and pointwise properness, although the latter property has not been so well studied in the literature. The basic theory of these properness conditions is summarized, with somewhat more attention paid to pointwise properness. C*-characterizations of the properties are proved, and applications to C*-dynamical systems are examined. This paper is partially expository, but some of the results are believed to be new.

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