arXiv subjects

Valentin Deaconu

Publications and source records attributed to Valentin Deaconu.

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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arXiv2024

Cohomology of ample groupoids

We introduce a cochain complex for ample groupoids $\mathcal G$ using a flat resolution defining their homology with coefficients in $\mathbb Z$. We prove that the cohomology of this cochain complex with values in a $\mathcal G$-module $M$ coincides with the previously introduced continuous cocycle cohomology of $\mathcal G$. In particular, this groupoid cohomology is invariant under Morita equivalence. We derive an exact sequence for the cohomology of skew products by a $\mathbb Z$-valued cocycle. We indicate how to compute the cohomology with coefficients in a $\mathcal G$-module $M$ for $AF$-groupoids and for certain action groupoids.

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arXiv2022

Groupoid actions and Koopman representations

We study the $C^*$-algebra $C^*(\kappa)$ generated by the Koopman representation $\kappa=\kappa^\mu$ of a locally compact groupoid $G$ acting on a measure space $(X,\mu)$, where $\mu$ is quasi-invariant for the action. We interpret $\kappa$ as an induced representation and we prove that if the groupoid $G\ltimes X$ is amenable, then $\kappa$ is weakly contained in the regular representation $\rho=\rho^\mu$ associated to $\mu$, so we have a surjective homomorphism $C^*_r(G)\to C^*(\kappa)$. We consider the particular case of Renault-Deaconu groupoids $G= G(X,T)$ acting on their unit space $X$ and show that in some cases $C^*(\kappa)\cong C^*(G)$.

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arXiv2022

Group Actions on Product Systems

We introduce the concept of crossed product of a product system by a locally compact group. We prove that the crossed product of a row-finite and faithful product system by an amenable group is also a row-finite and faithful product system. We illustrate with examples related to group actions on $k$-graphs and to higher rank Doplicher-Roberts algebras.

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arXiv2021

The Koopman representation for self-similar groupoid actions

We introduce the $C^*$-algebra $C^*(\kappa)$ generated by the Koopman representation $\kappa$ of an \'etale groupoid $G$ acting on a measure space $(X,\mu)$. We prove that for a level transitive self-similar action $(G,E)$ with $E$ finite and $|uE^1|$ constant, there is an invariant measure $\nu$ on $X=E^\infty$ and that $C^*(\kappa)$ is residually finite-dimensional with a normalized self-similar trace. We also discus $p$-fold similarities of Hilbert spaces in connection to representations of the graph algebra $C^*(E)$ and self-similar representations of $G$ in connection to the Cuntz-Pimsner algebra $C^*(G,E)$.

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arXiv2021

Higman-Thompson groups from self-similar groupoid actions

Given a self-similar groupoid action $(G,E)$ on a finite directed graph, we prove some properties of the corresponding ample groupoid of germs $\mathcal G(G,E)$. We study the analogue of the Higman-Thompson group associated to $(G,E)$ using $G$-tables and relate it to the topological full group of $\mathcal G(G,E)$, which is isomorphic to a subgroup of unitaries in the algebra $C^*(G,E)$. After recalling some concepts in groupoid homology, we discuss the Matui's AH-conjecture for $\mathcal G(G,E)$ in some particular cases.

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arXiv2020

On groupoids and $C^*$-algebras from self-similar actions

Given a self-similar groupoid action $(G,E)$ on the path space of a finite graph, we study the associated Exel-Pardo étale groupoid ${\mathcal G}(G,E)$ and its $C^*$-algebra $C^*(G,E)$. We review some facts about groupoid actions, skew products and semi-direct products and generalize a result of Renault about similarity of groupoids which resembles Takai duality. We also describe a general strategy to compute the $K$-theory of $C^*(G,E)$ and the homology of ${\mathcal G}(G,E)$ in certain cases and illustrate with an example.

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arXiv2020

C^*-algebras from k group representations

We introduce certain $C^*$-algebras and $k$-graphs associated to $k$ finite dimensional unitary representations $ρ_1,...,ρ_k$ of a compact group $G$. We define a higher rank Doplicher-Roberts algebra $\mathcal{O}_{ρ_1,...,ρ_k}$, constructed from intertwiners of tensor powers of these representations. Under certain conditions, we show that this $C^*$-algebra is isomorphic to a corner in the $C^*$-algebra of a row finite rank $k$ graph $Λ$ with no sources. For $G$ finite and $ρ_i$ faithful of dimension at least $2$, this graph is irreducible, it has vertices $\hat{G}$ and the edges are determined by $k$ commuting matrices obtained from the character table of the group. We illustrate with some examples when $\mathcal{O}_{ρ_1,...,ρ_k}$ is simple and purely infinite, and with some $K$-theory computations.

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arXiv2019

Symmetries of the C*-algebra of a vector bundle

We consider $C^*$-algebras constructed from compact group actions on complex vector bundles $E\to X$ endowed with a Hermitian metric. An action of $G$ by isometries on $E\to X$ induces an action on the $C^*$-correspondence $Γ(E)$ over $C(X)$ consisting of continuous sections, and on the associated Cuntz-Pimsner algebra $\mathcal O_E$, so we can study the crossed product $\mathcal O_E\rtimes G$. If the action is free and rank $E=n$, then we prove that $\mathcal O_E\rtimes G$ is Morita-Rieffel equivalent to a field of Cuntz algebras $\mathcal O_n$ over the orbit space $X/G$. If the action is fiberwise, then $\mathcal O_E\rtimes G$ becomes a continuous field of crossed products $\mathcal O_n\rtimes G$. For transitive actions, we show that $\mathcal O_E\rtimes G$ is Morita-Rieffel equivalent to a graph $C^*$-algebra.

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arXiv2017

Groupoid actions on $C^*$-correspondences

Let the groupoid $G$ with unit space $G^0$ act via a representation $ρ$ on a $C^*$-correspondence ${\mathcal H}$ over the $C_0(G^0)$-algebra $A$. By the universal property, $G$ acts on the Cuntz-Pimsner algebra ${\mathcal O}_{\mathcal H}$ which becomes a $C_0(G^0)$-algebra. The action of $G$ commutes with the gauge action on ${\mathcal O}_{\mathcal H}$, therefore $G$ acts also on the core algebra ${\mathcal O}_{\mathcal H}^{\mathbb T}$. We study the crossed product ${\mathcal O}_{\mathcal H}\rtimes G$ and the fixed point algebra ${\mathcal O}_{\mathcal H}^G$ and obtain similar results as in \cite{D}, where $G$ was a group. Under certain conditions, we prove that ${\mathcal O}_{\mathcal H}\rtimes G\cong {\mathcal O}_{\mathcal H\rtimes G}$, where $\mathcal H\rtimes G$ is the crossed product $C^*$-correspondence and that ${\mathcal O}_{\mathcal H}^G\cong{\mathcal O}_ρ$, where ${\mathcal O}_ρ$ is the Doplicher-Roberts algebra defined using intertwiners. The motivation of this paper comes from groupoid actions on graphs. Suppose $G$ with compact isotropy acts on a discrete locally finite graph $E$ with no sources. Since $C^*(G)$ is strongly Morita equivalent to a commutative $C^*$-algebra, we prove that the crossed product $C^*(E)\rtimes G$ is stably isomorphic to a graph algebra. We illustrate with some examples.

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arXiv2016

Cuntz-Pimsner Algebras of Group Representations

Given a locally compact group $G$ and a unitary representation $ρ:G\to U({\mathcal H})$ on a Hilbert space ${\mathcal H}$, we construct a $C^*$-correspondence ${\mathcal E}(ρ)={\mathcal H}\otimes_{\mathbb C} C^*(G)$ over $C^*(G)$ and study the Cuntz-Pimsner algebra ${\mathcal O}_{{\mathcal E}(ρ)}$. We prove that for $G$ compact, ${\mathcal O}_{{\mathcal E}(ρ)}$ is strong Morita equivalent to a graph $C^*$-algebra. If $λ$ is the left regular representation of an infinite, discrete and amenable group $G$, we show that ${\mathcal O}_{{\mathcal E}(λ)}$ is simple and purely infinite, with the same $K$-theory as $C^*(G)$. If $G$ is compact abelian, any representation decomposes into characters and determines a skew product graph. We illustrate with several examples and we compare ${\mathcal E}(ρ)$ with the crossed product $C^*$-correspondence.

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arXiv2016

Group actions on graphs and $C^*$-correspondences

If $G$ acts on a $C^*$-correspondence ${\mathcal H}$, then by the universal property $G$ acts on the Cuntz-Pimsner algebra ${\mathcal O}_{\mathcal H}$ and we study the crossed product ${\mathcal O}_{\mathcal H}\rtimes G$ and the fixed point algebra ${\mathcal O}_{\mathcal H}^G$. Using intertwiners, we define the Doplicher-Roberts algebra ${\mathcal O}_ρ$ of a representation $ρ$ of a compact group $G$ on ${\mathcal H}$ and prove that ${\mathcal O}_{\mathcal H}^G$ is isomorphic to ${\mathcal O}_ρ$. When the action of $G$ commutes with the gauge action on ${\mathcal O}_{\mathcal H}$, then $G$ acts also on the core algebras ${\mathcal O}_{\mathcal H}^{\mathbb T}$, where $\mathbb T$ denotes the unit circle. We give applications for the action of a group $G$ on the $C^*$-correspondence ${\mathcal H}_E$ associated to a directed graph $E$. If $G$ is finite and $E$ is discrete and locally finite, we prove that the crossed product $C^*(E)\rtimes G$ is isomorphic to the $C^*$-algebra of a graph of $C^*$-correspondences and stably isomorphic to a locally finite graph algebra. If $C^*(E)$ is simple and purely infinite and the action of $G$ is outer, then $C^*(E)^G$ and $C^*(E)\rtimes G$ are also simple and purely infinite with the same $K$-theory groups. We illustrate with several examples.

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arXiv2014

Crossed products and twisted $k$-graph algebras

An automorphism $β$ of a $k$-graph $Λ$ induces a crossed product $C^* ( Λ) \rtimes_β\mathbb{Z}$ which is isomorphic to a $(k+1)$-graph algebra $C^* ( Λ\times_β\mathbb{Z})$. In this paper we show how this process interacts with $k$-graph $C^*$-algebras which have been twisted by an element of their second cohomology group. This analysis is done using a long exact sequence in cohomology associated to this data. We conclude with some examples

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arXiv2011

Group actions on topological graphs

We define the action of a locally compact group $G$ on a topological graph $E$. This action induces a natural action of $G$ on the $C^*$-correspondence ${\mathcal H}(E)$ and on the graph $C^*$-algebra $C^*(E)$. If the action is free and proper, we prove that $C^*(E)\rtimes_r G$ is strongly Morita equivalent to $C^*(E/G)$. We define the skew product of a locally compact group $G$ by a topological graph $E$ via a cocycle $c:E^1\to G$. The group acts freely and properly on this new topological graph $E\times_cG$. If $G$ is abelian, there is a dual action on $C^*(E)$ such that $C^*(E)\rtimes \hat{G}\cong C^*(E\times_cG)$. We also define the fundamental group and the universal covering of a topological graph.

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arXiv2009

$C^*$-algebras and Fell bundles associated to a textile system

The notion of textile system was introduced by M. Nasu in order to analyze endomorphisms and automorphisms of topological Markov shifts. A textile system is given by two finite directed graphs $G$ and $H$ and two morphisms $p,q:G\to H$, with some extra properties. It turns out that a textile system determines a first quadrant two-dimensional shift of finite type, via a collection of Wang tiles, and conversely, any such shift is conjugate to a textile shift. In the case the morphisms $p$ and $q$ have the path lifting property, we prove that they induce groupoid morphisms $π, ρ:Γ(G)\to Γ(H)$ between the corresponding étale groupoids of $G$ and $H$. We define two families ${\mathcal A}(m,n)$ and $\bar{\mathcal A}(m,n)$ of $C^*$-algebras associated to a textile shift, and compute them in specific cases. These are graph algebras, associated to some one-dimensional shifts of finite type constructed from the textile shift. Under extra hypotheses, we also define two families of Fell bundles which encode the complexity of these two-dimensional shifts. We consider several classes of examples of textile shifts, including the full shift, the Golden Mean shift and shifts associated to rank two graphs.

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arXiv2009

Graphs of $C^*$-correspondences and Fell bundles

We define the notion of a $Λ$-system of $C^*$-correspondences associated to a higher-rank graph $Λ$. Roughly speaking, such a system assigns to each vertex of $Λ$ a $C^*$-algebra, and to each path in $Λ$ a $C^*$-correspondence in a way which carries compositions of paths to balanced tensor products of $C^*$-correspondences. Under some simplifying assumptions, we use Fowler's technology of Cuntz-Pimsner algebras for product systems of $C^*$-correspondences to associate a $C^*$-algebra to each $Λ$-system. We then construct a Fell bundle over the path groupoid $\Gg_Λ$ and show that the $C^*$-algebra of the $Λ$-system coincides with the reduced cross-sectional algebra of the Fell bundle. We conclude by discussing several examples of our construction arising in the literature.

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arXiv2009

Entropy of shifts on topological graph $C^*$-algebras

We give entropy estimates for two canonical non commutative shifts on $C^*$-algebras associated to some topological graphs $E=(E^0,E^1,s,r)$, defined using a basis of the corresponding Hilbert bimodule $H(E)$. We compare their entropies with the growth entropies associated directly to the topological graph. We illustrate with some examples of topological graphs considered by Katsura, where the vertex and the edge spaces are a union of unit circles and more detailed computations can be done.

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arXiv2007

Fell bundles associated to groupoid morphisms

Given a continuous open surjective morphism $π:G\to H$ of étale groupoids with amenable kernel, we construct a Fell bundle $E$ over $H$ and prove that its C*-algebra $C^*_r(E)$ is isomorphic to $C^*_r(G)$. This is related to results of Fell concerning C*-algebraic bundles over groups. The case $H=X$, a locally compact space, was treated earlier by Ramazan. We conclude that $C^*_r(G)$ is strongly Morita equivalent to a crossed product, the C*-algebra of a Fell bundle arising from an action of the groupoid $H$ on a C*-bundle over $H^0$. We apply the theory to groupoid morphisms obtained from extensions of dynamical systems and from morphisms of directed graphs with the path lifting property. We also prove a structure theorem for abelian Fell bundles.

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