arXiv subjects

Ruy Exel

Publications and source records attributed to Ruy Exel.

At least 19 recordsLinked to original sources

arXiv2026

Dynamics in large scale geometry

We investigate the large scale geometry of certain metric spaces through the lens of dynamics. Our approach establishes a close connection between large scale dynamical phenomena and operator algebras by characterizing various large scale dynamic behaviors in terms of GNS representations of the uniform Roe algebras arising from natural canonical states. Our dynamical systems are given by the Stone-\v{C}ech boundary of metric spaces together with their inverse semigroup of partial translations. This defines a space of orbits and we characterize Hausdorffness and $T_1$-ness of this space by the failure of coarse embeddability of certain metric spaces. Surprisingly, while the orbit space has very weak separation properties, we show that it satisfies a certain ''localized version'' of Urysohn's lemma. We show that the topology of the space of orbits and quasi-orbits are given by the space of irreducible representations of uniform Roe algebras and by the space of their primitive ideals, respectively. As a highlight of the theory developed herein, we provide classes of spaces such that the prime ideals of their uniform Roe algebras are primitive. This is the case for instance of spaces whose orbit space is $T_1$.

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arXiv2025

Consonant inverse semigroups

We study necessary and sufficient conditions for two inverse semigroups to possess identical tight groupoids from the point of view of their algebraic, topological, and spectral order structures. The spectral order is a partial order relation that presents itself very naturally on the tight groupoid of an inverse semigroup and is related to the subtle difference between tight filters and ultra-filters. Up to a small glitch, the spectral order makes tight groupoids into ordered groupoids in Ehresmann's sense. We introduce the notions of tight injectivity and tight surjectivity for inverse semigroup homomorphisms and show that, together, they provide necessary and sufficient conditions for the induced map to be an isomorphism of ordered topological groupoids. A homomorphism is then called a consonance provided these conditions are met. A consonance is not necessarily injective or surjective, so it is likely not an invertible map. Due to this fact, the existence of a consonance between inverse semigroups does not define an equivalence relation, so we consider instead the equivalence relation it generates. When it applies to a pair of inverse semigroups we say that they are consonant. The first main result of the paper shows that two inverse semigroups $S_1$ and $S_2$ are consonant if and only if their tight groupoids are isomorphic as ordered topological groupoids, if and only if there exists another inverse semigroup $T$ and consonances from each $S_i$ to $T$. We also prove that, given an inverse semigroup $S$, there exists a largest inverse semigroup consonant to $S$, denoted $S^\tau$, and called the tight envelope of $S$. In a nutshell, $S^\tau$ is the inverse semigroup formed by the compact up-slices (i.e.~slices that are up-sets relative to the spectral order) in the tight groupoid of $S$. Among other thi ... (cut short by the system)

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arXiv2024

Flows on uniform Roe algebras

For a uniformly locally finite metric space $(X, d)$, we investigate \emph{coarse} flows on its uniform Roe algebra $\mathrm{C}^*_u(X)$, defined as one-parameter groups of automorphisms whose differentiable elements include all partial isometries arising from partial translations on $X$. We first show that any flow $\sigma$ on $\mathrm{C}^*_u(X)$ corresponds to a (possibly unbounded) self-adjoint operator $h$ on $\ell_2(X)$ such that $\sigma_t(a) = e^{ith} a e^{-ith}$ for all $t \in \mathbb{R}$, allowing us to focus on operators $h$ that generate flows on $ \mathrm{C}^*_u (X)$. Assuming Yu's property A, we prove that a self-adjoint operator $h$ on $\ell_2(X)$ induces a coarse flow on $\mathrm{C}^*_u(X)$ if and only if $h$ can be expressed as $h = a + d$, where $a \in \mathrm{C}^*_u(X)$ and $d$ is a diagonal operator with entries forming a coarse function on $X$. We further study cocycle equivalence and cocycle perturbations of coarse flows, showing that, under property A, any coarse flow is a cocycle perturbation of a diagonal flow. Finally, for self-adjoint operators $h$ and $k$ that induce coarse flows on $\mathrm{C}^*_u(X)$, we characterize conditions under which the associated flows are either cocycle perturbations of each other or cocycle conjugate. In particular, if $h - k$ is bounded, then the flow induced by $h$ is a cocycle perturbation of the flow induced by $k$.

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arXiv2023

KMS states on uniform Roe algebras

We initiate the treatment of KMS states on uniform Roe algebras $\mathrm{C}^*_u(X)$ for a class of naturally occurring flows on these algebras. We show that KMS states on $\mathrm{C}^*_u(X)$ always factor through the diagonal operators $\ell_\infty(X)$. We show the study of those states splits into understanding their strongly continuous KMS states and the KMS states which vanish on the ideal of compact operators. We show strongly continuous states are always unique when they exist and we give explicit formulas for them. We link the study of KMS states which vanish on the compacts to the Higson corona of $X$ and provide lower bounds for the cardinality of the set of extreme KMS states. Lastly, we apply our theory to the $n$-branching tree.

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arXiv2023

Regular ideals under the ideal intersection property

The goal of this short note is to prove that when $A$ is a closed *-subalgebra of a C*-algebra $B$ satisfying the ideal intersection property plus a mild axiom (INV), then the map $J\mapsto J\cap A$ establishes an isomorphism from the boolean algebra of all regular ideals of $B$ to the boolean algebra of all regular, invariant ideals of $A$.

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arXiv2021

On Kumjian's C*-diagonals and the opaque ideal

We characterize exotic C*-algebras of twisted, principal \'etale groupoids, together with the abelian subalgebra associated to the unit space, as precisely being the inclusions "$A\subseteq B$" of C*-algebras in which $A$ is abelian, regular, and satisfies the extension property (pure states extend uniquely to $B$). When $B$ is moreover nuclear, we deduce that the corresponding opaque ideal is trivial. As an application, we give a streamlined characterization of Kumjian's C*-diagonals as the regular abelian subalgebras satisfying the extension property with vanishing opaque ideal.

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arXiv2021

Exotic Ideals in Free Transformation Group $C^*$-Algebras

Let $\Gamma$ be a discrete group acting freely via homeomorphisms on the compact Hausdorff space $X$ and let $C(X) \rtimes_\eta \Gamma$ be the completion of the convolution algebra $C_c(\Gamma,C(X))$ with respect to a $C^*$-norm $\eta$. A non-zero ideal $J \unlhd C(X) \rtimes_\eta \Gamma$ is exotic if $J \cap C(X) = \{0\}$. We show that exotic ideals are present whenever $\Gamma$ is non-amenable and there is an invariant probability measure on $X$. This fact, along with the recent theory of exotic crossed product functors, allows us to provide answers to two questions of K. Thomsen. Using the Koopman representation and a recent theorem of Elek, we show that when $\Gamma$ is a countably-infinite group having property (T) and $X$ is the Cantor set, there exists a free and minimal action of $\Gamma$ on $X$ and a $C^*$-norm $\eta$ on $C_c(\Gamma, C(X))$ such that $C(X)\rtimes_\eta\Gamma$ contains the compact operators as an exotic ideal. We use this example to provide a positive answer to a question of A. Katavolos and V. Paulsen. The opaque and grey ideals in $C(X)\rtimes_\eta \Gamma$ have trivial intersection with $C(X)$, and a result from arXiv:1901.09683 shows they coincide when the action of $\Gamma$ is free, however the problem of whether these ideals can be non-zero was left unresolved. We present an example of a free action of $\Gamma$ on a compact Hausdorff space $X$ along with a $C^*$-norm $\eta$ for which these ideals are non-trivial, in particular, they are exotic ideals.

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arXiv2019

Intermediate C*-algebras of Cartan Embeddings

Let $A$ be a C$^*$-algebra and let $D$ be a Cartan subalgebra of $A$. We study the following question: if $B$ is a C$^*$-algebra such that $D \subseteq B \subseteq A$, is $D$ a Cartan subalgebra of $B$? We give a positive answer in two cases: the case when there is a faithful conditional expectation from $A$ onto $B$, and the case when $A$ is nuclear and $D$ is a C$^*$-diagonal of $A$. In both cases there is a one-to-one correspondence between the intermediate C$^*$-algebras $B$, and a class of open subgroupoids of the groupoid $G$, where $Σ\rightarrow G$ is the twist associated with the embedding $D \subseteq A$.

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arXiv2019

Simplicity of algebras associated to non-Hausdorff groupoids

We prove a uniqueness theorem and give a characterization of simplicity for Steinberg algebras associated to non-Hausdorff ample groupoids. We also prove a uniqueness theorem and give a characterization of simplicity for the C*-algebra associated to non-Hausdorff étale groupoids. Then we show how our results apply in the setting of tight representations of inverse semigroups, groups acting on graphs, and self-similar actions. In particular, we show that C*-algebra and the complex Steinberg algebra of the self-similar action of the Grigorchuk group are simple but the Steinberg algebra with coefficients in $\mathbb{Z}_2$ is not simple.

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arXiv2019

Tight and cover-to-join representations of semilattices and inverse semigroups

We discuss the relationship between tight and cover-to-join representations of semilattices and inverse semigroups, showing that a slight extension of the former, together with an appropriate selection of co-domains, makes the two notions equivalent. As a consequence, when constructing universal objects based on them, one is allowed to substitute cover-to-join for tight and vice-versa.

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arXiv2018

Thermodynamic Formalism for Generalized Markov Shifts on Infinitely Many States

Given a 0-1 infinite matrix $A$ and its countable Markov shift $\Sigma_A$, one of the authors and M. Laca have introduced a kind of {\it generalized countable Markov shift} $X_A=\Sigma_A \cup Y_A$, where $Y_A$ is a special set of finite admissible words. For some of the most studied countable Markov shifts $\Sigma_A$, $X_A$ is a compactification of $\Sigma_A$, and always it is at least locally compact. We developed the thermodynamic formalism on the space $X_A$, exploring the connections with standard results on $\Sigma_A$. New phenomena appear, such as new conformal measures and a {\it length-type phase transition}: the eigenmeasure lives on $\Sigma_A$ at high temperature and lives on $Y_A$ at low temperature. Using a pressure-point definition proposed by M. Denker and M. Yuri for iterated function systems, we proved that the Gurevich pressure is a natural definition for the pressure function in the generalized setting. For the gauge action, the Gurevich entropy is a critical temperature for the existence of new conformal measures (KMS states) living on $Y_A$. We exhibit examples with infinitely (even uncountable) many new extremal conformal measures, undetectable in the usual formalism. We prove that conformal measures always exist at low temperatures when the potential is coercive enough. We characterized a basis of the topology of $X_A$ to study the weak$^*$ convergence of measures on $X_A$, and we show some cases where the conformal measure living on $Y_A$ converges to a conformal one living on $\Sigma_A$. We prove the equivalence among several notions of conformality for locally compact Hausdorff second countable spaces, including quasi-invariant measures for generalized Renault-Deaconu groupoids.

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arXiv2018

$C^*$-algebras of self-similar graphs over arbitrary graphs

In this note we extend the construction of a $C^*$-algebra associated to a self-similar graph to the case of arbitrary countable graphs. We reduce the problem to the row-finite case with no sources, by using a desingularization process. Finally, we characterize simplicity in this case.

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arXiv2017

Partial Dynamical Systems, Fell Bundles and Applications

This is a book about Partial Actions and Fell Bundles with applications to C*-algebras generated by partial isometries. Here is the table of contents: 1-Introduction, 2-Partial actions, 3-Restriction and globalization, 4-Inverse semigroups, 5-Topological partial dynamical systems, 6-Algebraic partial dynamical systems, 7-Multipliers, 8-Crossed products, 9-Partial group representations, 10-Partial group algebras, 11-C*-algebraic partial dynamical systems, 12-Partial isometries, 13-Covariant representations of C*-algebraic dynamical systems, 14-Partial representations subject to relations, 15-Hilbert modules and Morita-Rieffel-equivalence, 16-Fell bundles, 17-Reduced cross-sectional algebras, 18-Fell's absorption principle, 19-Graded C*-algebras, 20-Amenability for Fell bundles, 21-Functoriality for Fell bundles, 22-Functoriality for partial actions, 23-Ideals in graded algebras, 24-Pre-Fell-bundles, 25-Tensor products of Fell bundles, 26-Smash product, 27-Stable Fell bundles as partial crossed products, 28-Globalization in the C*-context, 29-Topologically free partial actions, 30-Dilating partial representations, 31-Semigroups of isometries, 32-Quasi-lattice ordered groups, 33-C*-algebras generated by semigroups of isometries, 34-Wiener-Hopf C*-algebras, 35-The Toeplitz C*-algebra of a graph, 36-Path spaces, 37-Graph C*-algebras, 38-References, 39-Subject Index.

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arXiv2017

Reduced C*-algebras of Fell bundles over inverse semigroups

We construct a weak conditional expectation from the section C*-algebra of a Fell bundle over a unital inverse semigroup to its unit fibre. We use this to define the reduced C*-algebra of the Fell bundle. We study when the reduced C*-algebra for an inverse semigroup action on a groupoid by partial equivalences coincides with the reduced groupoid C*-algebra of the transformation groupoid, giving both positive results and counterexamples.

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arXiv2016

Self-similar graphs, a unified treatment of Katsura and Nekrashevych C*-algebras

Given a graph $E$, an action of a group $G$ on $E$, and a $G$-valued cocycle $ϕ$ on the edges of $E$, we define a C*-algebra denoted ${\cal O}_{G,E}$, which is shown to be isomorphic to the tight C*-algebra associated to a certain inverse semigroup $S_{G,E}$ built naturally from the triple $(G,E,ϕ)$. As a tight C*-algebra, ${\cal O}_{G,E}$ is also isomorphic to the full C*-algebra of a naturally occurring groupoid ${\cal G}_{tight}(S_{G,E})$. We then study the relationship between properties of the action, of the groupoid and of the C*-algebra, with an emphasis on situations in which ${\cal O}_{G,E}$ is a Kirchberg algebra. Our main applications are to Katsura algebras and to certain algebras constructed by Nekrashevych from self-similar groups. These two classes of C*-algebras are shown to be special cases of our ${\cal O}_{G,E}$, and many of their known properties are shown to follow from our general theory.

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arXiv2016

A Generalised uniqueness theorem and the graded ideal structure of Steinberg algebras

Given an ample, Hausdorff groupoid $\mathcal{G}$, and a unital commutative ring $R$, we consider the Steinberg algebra $A_R(\mathcal {G})$. First we prove a uniqueness theorem for this algebra and then, when $\mathcal{G}$ is graded by a cocycle, we study graded ideals in $A_R(\mathcal {G})$. Applications are given for two classes of ample groupoids, namely those coming from actions of groups on graphs, and also to groupoids defined in terms of Boolean dynamical systems.

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arXiv2016

Inverse semigroup actions as groupoid actions

To an inverse semigroup, we associate an étale groupoid such that its actions on topological spaces are equivalent to actions of the inverse semigroup. Both the object and the arrow space of this groupoid are non-Hausdorff. We show that this construction provides an adjoint functor to the functor that maps a groupoid to its inverse semigroup of bisections, where we turn étale groupoids into a category using algebraic morphisms. We also discuss how to recover a groupoid from this inverse semigroup.

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arXiv2015

Topological entropy for partial actions of the group $\mathbb Z$

In this paper we introduce the definition of entropy for a partial $\mathbb{Z}$-action. We show that the definition of partial entropy is an extension of the definition of topological entropy for a $\mathbb Z$-action. We also prove that the partial topological entropy is concentrated on the non-wandering set.

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