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Danilo Royer

Publications and source records attributed to Danilo Royer.

At least 19 recordsLinked to original sources
arXiv2026

Residual finite-dimensionality of ultragraph algebras via branching systems

We study residual finite-dimensionality for ultragraph algebras, both in the algebraic and in the C-star-algebraic settings. We introduce graph-theoretic RFD conditions for ultragraphs, extending the conditions that characterize RFD graph C-star-algebras. Using the boundary ultrapath branching system, we construct finite-dimensional branching-system representations associated to terminal boundary sets and no-exit cycles. These representations are used to prove that, whenever an ultragraph satisfies the graph-theoretic RFD conditions, its ultragraph Leavitt path algebra LK(G) is RFD, for every field K, and its ultragraph C-star-algebra RFD. For ultragraphs satisfying Condition (RFUM2), we prove converses in both settings. The analytic converse uses the groupoid model and the density of periodic points, while the algebraic converse is proved directly by finite-dimensional linear algebra. Thus, for RFUM2 ultragraphs, RFD of LK(G), RFD of C(G), and the graph-theoretic RFD conditions are equivalent. This gives, in particular, a common combinatorial description linking the algebraic and analytic theories, recovers the graph C-start-algebra characterization, and yields an algebraic characterization for Leavitt path algebras of graphs. We also construct an RFD ultragraph algebra which is genuinely outside the graph-algebra class in both settings.

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arXiv2026

Paterson compactifications, inverse limits and shadowing for Deaconu-Renault systems

We develop a new metric and inverse-limit framework for Deaconu-Renault systems arising from local homeomorphisms between open subsets of locally compact zero-dimensional spaces. Our starting point is the Paterson-type compactification of infinite product spaces, which underlies several symbolic and groupoid models, including one-sided shifts over infinite alphabets and path spaces of graphs and higher-rank graphs. We construct an explicit compatible ultrametric on this compactification and give a concrete description of its generalized cylinder topology and convergence. Within this framework, we introduce an inverse-limit-type space naturally associated to a Deaconu-Renault system. In contrast with the classical compact theory, the correct inverse-limit object must incorporate not only infinite backward orbits but also finite configurations arising as limits of such orbits. This produces a canonical shift system extending the original dynamics. We then study shadowing in the ultrametric setting. For spaces admitting tame defining sequences, we characterize shadowing in terms of the defining partitions, extending the topological description of shadowing from compact zero-dimensional dynamics to the locally compact setting relevant for Deaconu-Renault systems. As a main application, we prove a transfer theorem showing, under a separation property expressed through uniformly contracting inverse branches, that shadowing for the inverse-limit Deaconu-Renault system is equivalent to shadowing for the compactified base system. This provides a noncompact inverse-limit shadowing theory for a broad class of partially defined local-homeomorphism dynamics.

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arXiv2025

Relative ultragraph algebras and infinite interval maps

In this paper, we introduce the notion of relative ultragraph algebras and extend classical injectivity criteria for representations, particularly those arising from branching systems,to this relative setting. This new concept is closely connected to, and indeed motivated by, the dynamics of certain interval maps that we call infinite Markov maps. The central point is that these algebras provide the natural framework in which representations induced by such maps are defined.

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arXiv2024

Nonwandering sets and the entropy of local homeomorphisms

A local homeomorphism between open subsets of a locally compact Hausdorff space induces dynamical systems with a wide range of applications, including in C*-algebras. In this paper, we introduce the concepts of nonwandering and wandering sets for such systems and show that, under mild conditions, the metric entropy is concentrated in the nonwandering set. More generally, we demonstrate that the entropy of the system is the maximum of the entropies of the systems restricted to the nonwandering set and the closure of the wandering set. We illustrate these results with several examples, including applications to subshifts over countable alphabets.

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arXiv2024

The socle of subshift algebras, with applications to subshift conjugacy

We introduce the concept of "irrational paths" for a given subshift and use it to characterize all minimal left ideals in the associated unital subshift algebra. Consequently, we characterize the socle as the sum of the ideals generated by irrational paths. Proceeding, we construct a graph such that the Leavitt path algebra of this graph is graded isomorphic to the socle. This realization allows us to show that the graded structure of the socle serves as an invariant for the conjugacy of Ott-Tomforde-Willis subshifts and for the isometric conjugacy of subshifts constructed with the product topology. Additionally, we establish that the socle of the unital subshift algebra is contained in the socle of the corresponding unital subshift C*-algebra.

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arXiv2023

Irreducible representations of one-sided subshift algebras

Given a subshift over an arbitrary alphabet, we construct a representation of the associated unital algebra. We describe a criteria for the faithfulness of this representation in terms of the existence of cycles with no exits. Subsequently, we focus on describing the irreducible components of this representation and characterize equivalence between such components. Building upon these findings, we identify the minimal left ideals of a subshift algebra associated with (what we call) line paths.

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arXiv2023

The reduction theorem for algebras of one-sided subshifts over arbitrary alphabets

Let $R$ be a commutative unital ring, $\textsf{X}$ a subshift, and $\widetilde{\mathcal{A}}_R(\textsf{X})$ the corresponding unital subshift algebra. We establish the reduction theorem for $\widetilde{\mathcal{A}}_R(\textsf{X})$. As a consequence, we obtain a Cuntz-Krieger uniqueness theorem for $\widetilde{\mathcal{A}}_R(\textsf{X})$ and we show that $\widetilde{\mathcal{A}}_R(\textsf{X})$ is semiprimitive (resp. semiprime) whenever $R$ is a field (resp. a domain).

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arXiv2023

Entropy of local homeomorphisms with applications to infinite alphabet shift spaces

In this paper, we introduce topological entropy for dynamical systems generated by a single local homeomorphism (Deaconu-Renault systems). More precisely, we generalize Adler, Konheim, and McAndrew's definition of entropy via covers and Bowen's definition of entropy via separated sets. We propose a definition of factor map between Deaconu-Renault systems and show that entropy (via separated sets) always decreases under uniformly continuous factor maps. Since the variational principle does not hold in the full generality of our setting, we show that the proposed entropy via covers is a lower bound to the proposed entropy via separated sets. Finally, we compute entropy for infinite graphs (and ultragraphs) and compare it with the entropy of infinite graphs defined by Gurevich.

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arXiv2021

Properties of the gradings on ultragraph algebras via the underlying combinatorics

There are two established gradings on Leavitt path algebras associated with ultragraphs, namely the grading by the integers group and the grading by the free group on the edges. In this paper, we characterize properties of these gradings in terms of the underlying combinatorial properties of the ultragraphs. More precisely, we characterize when the gradings are strong or epsilon-strong. The results regarding the free group on the edges are new also in the context of Leavitt path algebras of graphs. Finally, we also describe the relation between the strongness of the integer grading on an ultragraph Leavitt path algebra and the saturation of the gauge action associated with the corresponding ultragraph C*-algebra.

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arXiv2020

A note on the regular ideals of Leavitt path algebras

We show that, for an arbitrary graph, a regular ideal of the associated Leavitt path algebra is also graded. As a consequence, for a row-finite graph, we obtain that the quotient of the associated Leavitt path by a regular ideal is again a Leavitt path algebra and that Condition~(L) is preserved by quotients by regular ideals. Furthermore, we describe the vertex set of a regular ideal and make a comparison between the theory of regular ideals in Leavitt path algebras and in graph C*-algebras.

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arXiv2019

Representations of C*-algebras of row-countable graphs and unitary equivalence

In this article we show that there are branching systems (which induce representations of the graph algebra $C^*(E)$) associated to each row-countable graph $E$. For row-countable graphs, we characterize the condition $(L)$ via branching systems. Moreover, we show that each permutative representation in Hilbert spaces operators is unitarily equivalent to one induced by a branching system, even the spaces being not separable. Furthermore, under some hypothesis on the graph, we show that each representation of the graph C*-algebra is permutative.

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arXiv2019

Irreducible and permutative representations of ultragraph Leavitt path algebras

We completely characterize perfect, permutative, irreducible representations of an ultragraph Leavitt path algebra. For this we extend to ultragraph Leavitt path algebras Chen's construction of irreducible representations of Leavitt path algebras. We show that these representations can be built from branching system and characterize irreducible representations associated to perfect branching systems. Along the way we improve the characterization of faithfulness of Chen's irreducible representations.

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arXiv2019

Representations and the reduction theorem for ultragraph Leavitt path algebras

In this paper we study representations of ultragraph Leavitt path algebras via branching systems and, using partial skew ring theory, prove the reduction theorem for these algebras. We apply the reduction theorem to show that ultragraph Leavitt path algebras are semiprime and to completely describe faithfulness of the representations arising from branching systems, in terms of the dynamics of the branching systems. Furthermore, we study permutative representations and provide a sufficient criteria for a permutative representation of an ultragraph Leavitt path algebra to be equivalent to a representation arising from a branching system. We apply this criteria to describe a class of ultragraphs for which every representation (satisfying a mild condition) is permutative and has a restriction that is equivalent to a representation arising from a branching system.

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arXiv2018

Simplicity of skew inverse semigroup rings with applications to Steinberg algebras and topological dynamics

Given a partial action $π$ of an inverse semigroup $S$ on a ring $\mathcal{A}$ one may construct its associated skew inverse semigroup ring $\mathcal{A} \rtimes_πS$. Our main result asserts that, when $\mathcal{A}$ is commutative, the ring $\mathcal{A} \rtimes_πS$ is simple if, and only if, $\mathcal{A}$ is a maximal commutative subring of $\mathcal{A} \rtimes_πS$ and $\mathcal{A}$ is $S$-simple. We apply this result in the context of topological inverse semigroup actions to connect simplicity of the associated skew inverse semigroup ring with topological properties of the action. Furthermore, we use our result to present a new proof of the simplicity criterion for a Steinberg algebra $A_R(\mathcal{G})$ associated with a Hausdorff and ample groupoid $\mathcal{G}$.

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arXiv2018

Partial group algebra with projections and relations

We introduce the notion of the partial group algebra with projections and relations and show that this C*-algebra is a partial crossed product. Examples of partial group algebras with projections and relations are the Cuntz-Krieger algebras and the unitization of C*-algebras of directed graphs.

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arXiv2017

Infinite alphabet edge shift spaces via ultragraphs and their C*-algebras

We define a notion of (one-sided) edge shift spaces associated to ultragraphs. In the finite case our notion coincides with the edge shift space of a graph. In general, we show that our space is metrizable and has a countable basis of clopen sets. We show that for a large class of ultragraphs the basis elements of the topology are compact. We examine shift morphisms between these shift spaces, and, for the locally compact case, show that if two (possibly infinite) ultragraphs have edge shifts that are conjugate, via a conjugacy that preserves length, then the associated ultragraph C*-algebras are isomorphic. To prove this last result we realize the relevant ultragraph C*-algebras as partial crossed products.

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arXiv2017

Branching Systems for Higher-Rank Graph C*-algebras

We define branching systems for finitely aligned higher-rank graphs. From these we construct concrete representations of higher-rank graph C*-algebras on Hilbert spaces. We prove a generalized Cuntz-Krieger uniqueness theorem for periodic single-vertex 2-graphs. We use this result to give a sufficient condition under which representations of periodic single-vertex 2-graph C*-algebras arising from branching systems are faithful.

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