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S. Kaliszewski

Publications and source records attributed to S. Kaliszewski.

At least 19 recordsLinked to original sources
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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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

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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arXiv2019

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

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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