arXiv subjects

Gilles G. de Castro

Publications and source records attributed to Gilles G. de Castro.

At least 19 recordsLinked to original sources

arXiv2026

Pullbacks of Groupoid $C^*$-Algebras over Disjoint Invariant Sets

We establish a pullback theorem for $C^*$-algebras of locally compact Hausdorff étale groupoids. The theorem shows that, when the unit space is covered by two closed invariant subsets whose complements are disjoint open invariant subsets, the corresponding groupoid $C^*$-algebras form a pullback diagram in the category of $\mathbb T$-$C^*$-algebras and $\mathbb T$-equivariant $*$-homomorphisms, for the gauge actions induced by a $\mathbb Z$-valued cocycle and its restrictions. We then prove a collection of boundary-path decomposition theorems for graphs, relative graphs, and topological graphs. We show that admissible decompositions of graphs, together with their analogues in the relative and topological settings, induce corresponding decompositions of boundary path spaces. Combining these decomposition theorems with the groupoid pullback theorem, we recover previously known pullback theorems for graph $C^*$-algebras, relative graph $C^*$-algebras, and topological graph $C^*$-algebras. Thus these pullback phenomena are explained by a single groupoid-theoretic mechanism.

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arXiv2026

Notes on universal C*-algebras

In these notes, we explain in details the construction of the universal C*-algebra as done by Blackadar, starting with the construction of the free *-algebra. At the end, we explain the extension done by Boava and the author to include relations described using the strong operator topology. As an example, we show that a graph C*-algebra can be defined using infinite sums.

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arXiv2026

Relation morphisms of directed graphs

Associating graph algebras to directed graphs leads to both covariant and contravariant functors from suitable categories of graphs to the category k-Alg of algebras and algebra homomorphisms. As both functors are often used at the same time, finding a new category of graphs that allows a "common denominator" functor unifying the covariant and contravariant constructions is a fundamental problem. Herein, we solve this problem by first introducing the relation category of graphs RG, and then determining the concept of admissible graph relations that yields a subcategory of RG admitting a contravariant functor to k-Alg simultaneously generalizing the aforementioned covariant and contravariant functors. Although we focus on Leavitt path algebras and graph C*-algebras, on the way we unravel functors to k-Alg given by path algebras, Cohn path algebras and Toeplitz graph C*-algebras from suitable subcategories of RG. Better still, we illustrate relation morphisms of graphs by naturally occurring examples, including Cuntz algebras, quantum spheres and quantum balls.

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arXiv2026

Graded Locally Finite and Just Infinite Steinberg Algebras

We study graded locally finite and graded just infinite Steinberg algebras of ample Hausdorff groupoids. For gradings by discrete groups induced by a cocycle, we characterize finite-dimensional homogeneous components in terms of finite cocycle fibres. Moreover, we obtain general criteria ensuring that, once one homogeneous component is infinite-dimensional, all of them are. Under suitable isotropy hypotheses, we show that for locally finite Steinberg algebras all irreducible representations act by finite-rank operators. We also obtain necessary conditions for Steinberg algebras to be just infinite. In addition, we give a complete characterization of graded just infinite Steinberg algebras in terms of finite invariant reductions and identify conditions under which graded just infiniteness is equivalent to graded simplicity. We apply these results to Steinberg algebras of Deaconu--Renault groupoids and to subshift algebras. For Deaconu--Renault groupoids, we obtain dynamical criteria for graded just infiniteness. For subshift algebras, we show that there are no nontrivial finite-dimensional examples, we characterize when all homogeneous components are infinite-dimensional, and in the finite-alphabet case, we prove that graded just infiniteness, graded simplicity, minimality of the associated groupoid, and hyper-cofinality of the subshift are equivalent.

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arXiv2026

The functoriality of moves on graphs and the extended covariant functoriality of graph algebras

Combinatorics of graphs is a very powerful tool to unravel various properties of graph algebras. In particular, isomorphisms between graph algebras are often implemented by moves between their graphs. In this paper, we make these combinatorial methods functorial, and show that collapsing an out-split graph to the original graph and transforming a graph to a shifted graph can be implemented by admissible graph homomorphisms and admissible path homomorphisms, respectively. To include the inverses of such isomorphisms, we introduce a new category of graphs where morphisms are given as regular homomorphisms of graph inverse semigroups. This new category admits a covariant functor to the category of C*-algebras and $*$-homomorphisms which extends the known covariant functor from the category of graphs and admissible path homomorphisms.

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arXiv2025

Graph morphisms as groupoid actors

We describe proper actors from the underlying groupoid of a graph C*-algebra to another étale groupoid in terms of bisections. This allows to understand graph morphisms and the *-homomorphisms that they induce more conceptually. More generally, we describe actors from the groupoid model of a groupoid correspondence to any étale groupoid. This also covers the groupoids associated to self-similar groups and self-similar graphs, among others.

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arXiv2024

The dynamical structure of partial group algebras with relations, with applications to subshift algebras

We introduce partial group algebras with relations in a purely algebraic framework. Given a group and a set of relations, we define an algebraic partial action and prove that the resulting partial skew group ring is isomorphic to the associated partial group algebra with relations. Under suitable conditions - which always holds if the base ring is a field - we demonstrate that the partial skew group ring can also be described using a topological partial action. Furthermore, we show how subshift algebras can be realized as partial group algebras with relations. Using the topological partial action, we describe simplicity of subshift algebras in terms of the underlying dynamics of the subshift.

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arXiv2024

A Categorical Interpretation of Continuous Orbit Equivalence for Partial Dynamical Systems

We define the orbit morphism of partial dynamical systems and prove that an orbit morphism being an isomorphism in the category of partial dynamical systems and orbit morphisms is equivalent to the existence of a continuous orbit equivalence between the given partial dynamical systems that preserves the essential stabilisers. We show that this is equivalent to the existence of a diagonal-preserving isomorphism between the corresponding crossed products when the essential stabilisers of partial actions are torsion-free and abelian. We also characterize when an étale groupoid is isomorphic to the transformation groupoid of some partial action. Additionally, we explore the implications in the context of semi-saturated orthogonal partial dynamical systems over free groups, establishing connections with Deaconu-Renault systems and the concept of eventual conjugacy. Finally, we apply our results to C*-algebras associated with generalized Boolean dynamical systems.

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arXiv2024

Ideals of étale groupoid algebras with coefficients in a sheaf with applications to topological dynamics

We prove the Effros-Hahn conjecture for groupoid algebras with coefficients in a sheaf, obtaining as a consequence a description of the ideals in skew inverse semigroup rings. We also use the description of the ideals to characterize when the groupoid algebras with coefficients in a sheaf are von Neumann regular, primitive, semiprimitive, or simple. We apply our results to the topological dynamics of actions of inverse semigroups, describing the existence of dense orbits and minimality in terms of primitivity and simplicity, respectively, of the associated algebra. Moreover, we apply our results to the usual complex groupoid algebra of continuous functions with compact support, used to build the C*-algebra associated with a groupoid, and describe criteria for its simplicity.

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arXiv2024

Étale categories, restriction semigroups, and their operator algebras

We define the full and reduced non-self-adjoint operator algebras associated with étale categories and restriction semigroups, answering a question posed by Kudryavtseva and Lawson in \cite{lawson}. Moreover, we define the semicrossed product algebra of an étale action of a restriction semigroup on a $C^*$-algebra, which turns out to be the key point when connecting the operator algebra of a restriction semigroup with the operator algebra of its associated étale category. We also prove that in the particular cases of étale groupoids and inverse semigroups our operator algebras coincide with the $C^*$-algebras of the referred objects.

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arXiv2023

C*-Algebras of one-sided subshifts over arbitrary alphabets

We associate a C*-algebra $\widetilde{\mathcal{O}}_{\textsf{X}}$ with a subshift over an arbitrary, possibly infinite, alphabet. We show that $\widetilde{\mathcal{O}}_{\textsf{X}}$ is a full invariant for topological conjugacy of the subshifts of Ott, Tomforde, and Willis. When the alphabet is countable, we show that $\widetilde{\mathcal{O}}_{\textsf{X}}$ is an invariant for isometric conjugacy of subshifts with the product metric. For a suitable partial action associated with a subshift over a countable alphabet, we show that $\widetilde{\mathcal{O}}_{\textsf{X}}$ is also an invariant for continuous orbit equivalence. Additionally, we give a concrete way to compute the K-theory of $\widetilde{\mathcal{O}}_{\textsf{X}}$ and illustrate it with two examples.

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arXiv2023

Algebras of one-sided subshifts over arbitrary alphabets

We introduce two algebras associated with a subshift over an arbitrary alphabet. One is unital and the other not necessarily. We focus on the unital case and describe a conjugacy between Ott-Tomforde-Willis subshifts in terms of a homeomorphism between the Stone duals of suitable Boolean algebras, and in terms of a diagonal-preserving isomorphism of the associated unital algebras. For this, we realise the unital algebra associated with a subshift as a groupoid algebra and as a partial skew group ring.

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arXiv2022

C*-algebras of generalized Boolean dynamical systems as partial crossed products

In this paper, we realize C*-algebras of generalized Boolean dynamical systems as partial crossed products. Reciprocally, we give some sufficient conditions for a partial crossed product to be isomorphic to a C*-algebra of a generalized Boolean dynamical system. As an application, we show that gauge-invariant ideals of C*-algebras of generalized Boolean dynamical systems are themselves C*-algebras of generalized Boolean dynamical system.

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arXiv2021

Reconstruction of twisted Steinberg algebras

We show how to recover a discrete twist over an ample Hausdorff groupoid from a pair consisting of an algebra and what we call a quasi-Cartan subalgebra. We identify precisely which twists arise in this way (namely, those that satisfy the local bisection hypothesis), and we prove that the assignment of twisted Steinberg algebras to such twists and our construction of a twist from a quasi-Cartan pair are mutually inverse. We identify the algebraic pairs that correspond to effective groupoids and to principal groupoids. We also indicate the scope of our results by identifying large classes of twists for which the local bisection hypothesis holds automatically.

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arXiv2021

KMS states for generalized gauge actions on C*-algebras associated with self-similar sets

Given a self-similar $K$ set defined from an iterated function system $Γ=(γ_1,\ldots,γ_n)$ and a set of function $H=\{h_i:K\to\mathbb{R}\}_{i=1}^d$ satisfying suitable conditions, we define a generalized gauge action on Kawjiwara-Watatani algebras $\mathcal{O}_Γ$ and their Toeplitz extensions $\mathcal{T}_Γ$. We then characterize the KMS states for this action. For each $β\in(0,\infty)$, there is a Ruelle operator $\mathcal{L}_{H,β}$ and the existence of KMS states at inverse temperature $β$ is related to this operator. The critical inverse temperature $β_c$ is such that $\mathcal{L}_{H,β_c}$ has spectral radius 1. If $β<β_c$, there are no KMS states on $\mathcal{O}_Γ$ and $\mathcal{T}_Γ$; if $β=β_c$, there is a unique KMS state on $\mathcal{O}_Γ$ and $\mathcal{T}_Γ$ which is given by the eigenmeasure of $\mathcal{L}_{H,β_c}$; and if $β>β_c$, including $β=\infty$, the extreme points of the set of KMS states on $\mathcal{T}_Γ$ are parametrized by the elements of $K$ and on $\mathcal{O}_Γ$ by the set of branched points.

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arXiv2021

Boundary path groupoids of generalized Boolean dynamical systems and their C*-algebras

In this paper, we provide two types of boundary path groupoids from a generalized Boolean dynamical system $(\mathcal{B},\mathcal{L}, θ, \mathcal{I}_α)$. For the first groupoid, we associate an inverse semigroup to a generalized Boolean dynamical system and use the tight spectrum $\mathsf{T}$ as the unit space of a groupoid $Γ(\mathcal{B},\mathcal{L}, θ, \mathcal{I}_α)$ that is isomorphic to the tight groupoid $\mathcal{G}_{tight}$. The other one is defined as the Renault-Deaconu groupoid $Γ(\partial E, σ_E)$ arising from a topological correspondence $E$ associated with a generalized Boolean dynamical system. We then prove that the tight spectrum $\mathsf{T} $ is homeomorphic to the boundary path space $\partial E$ obtained from the topological correspondence. Using this result, we prove that the groupoid $Γ(\mathcal{B},\mathcal{L}, θ, \mathcal{I}_α)$ equipped with the topology induced from the topology on $\mathcal{G}_{tight}$ is isomorphic to $Γ(\partial E, σ_E)$ as a topological groupoid. Finally, we show that their $C^*$-algebras are isomorphic to the $C^*$-algebra of the generalized Boolean dynamical system.

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arXiv2021

Leavitt path algebras of labelled graphs

A Leavitt labelled path algebra over a commutative unital ring is associated with a labelled space, generalizing Leavitt path algebras associated with graphs and ultragraphs as well as torsion-free commutative algebras generated by idempotents. We show that Leavitt labelled path algebras can be realized as partial skew group rings, Steinberg algebras, and Cuntz-Pimsner algebras. Via these realizations we obtain generalized uniqueness theorems, a description of diagonal preserving isomorphisms and we characterize simplicity of Leavitt labelled path algebras. In addition, we prove that a large class of partial skew group rings can be realized as Leavitt labelled path algebras.

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arXiv2020

Ultragraph algebras via labelled graph groupoids, with applications to generalized uniqueness theorems

An ultragraph gives rise to a labelled graph with some particular properties. In this paper we describe the algebras associated to such labelled graphs as groupoid algebras. More precisely, we show that the known groupoid algebra realization of ultragraph C*-algebras is only valid for ultragraphs for which the range of each edge is finite, and we extend this realization to any ultragraph (including ultragraphs with sinks). Using our machinery, we characterize the shift space associated to an ultragraph as the tight spectrum of the inverse semigroup associated to the ultragraph (viewed as a labelled graph). Furthermore, in the purely algebraic setting, we show that the algebraic partial action used to describe an ultragraph Leavitt path algebra as a partial skew group ring is equivalent to the dual of a topological partial action, and we use this to describe ultragraph Leavitt path algebras as Steinberg algebras. Finally, we prove generalized uniqueness theorems for both ultragraph C*-algebras and ultragraph Leavitt path algebras and characterize their abelian core subalgebras.

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