arXiv subjects

Pere Ara

Publications and source records attributed to Pere Ara.

At least 19 recordsLinked to original sources

arXiv2026

Dynamics on graphs with disjoint cycles and applications

In this article, we introduce the notion of connected finite graphs with disjoint cycles in normal form and show that any such graph can be transformed into a normal form graph via a finite sequence of in-splittings and out-splittings. Consequently, we provide number-theoretic criteria for meteor graphs of length three to be strongly shift equivalent, where a meteor graph of length three is a connected finite essential graph consisting of three disjoint cycles which makes a unique chain of cycles of length three. We then prove that meteor graphs of length three whose cycle lengths are pairwise coprime are shift equivalent if and only if they are strongly shift equivalent, if and only if their corresponding Leavitt path algebras are graded Morita equivalent, if and only if their graded $K$-theories, $K^{gr}_0$, are order-preserving $\mathbb{Z}[x, x^{-1}]$-module isomorphic. As a consequence, Williams' Conjecture and Hazrat's Graded Morita Equivalence Conjecture hold for graphs with disjoint cycles that contain exactly three cycles whose lengths are pairwise coprime.

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arXiv2026

Positive definite matrices and involutions: the manners of their infinite cousins

Matrices and involutions in/on the algebras $B(\mathbb{R})$ and $B(\mathbb{C})$ of real (resp., complex) row- and column-finite $\omega\times\omega$ matrices are studied. It is proved that any positive definite $\mathbb{R}$-algebra involution on $B(\mathbb{R})$ (resp., any positive definite conjugate-linear involution on $B(\mathbb{C})$) is given by conjugating the transpose involution (resp., the conjugate-transpose involution) with a positive definite matrix from the algebra in question. All positive definite matrices in these algebras have Cholesky factorizations within the algebra. Examples are constructed to show that such positive definite matrices need not have any eigenvalues for their canonical action on column-finite column vectors, and they need not have any square roots in $B(\mathbb{C})$.

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arXiv2026

An introduction to separated graphs and their type semigroups

We introduce $C^*$-algebras associated with directed graphs, along with two generalizations of this concept, namely Exel-Pardo $C^*$-algebras associated with a self-similar action of a group on a directed graph, and the $C^*$-algebras associated with separated graphs. These constructions have in common that they have a dynamical behavior, being the groupoid $C^*$-algebras associated to certain topological groupoids, which are built from the combinatorial structure. An important invariant one may associate to these dynamical systems is the so-called type semigroup. We will find a formula to compute the type semigroup for a general self-similar action of a group on a row-finite graph $E$ without sources, following a recent paper by Kwa\'sniewski, Meyer and Prasad, and for any finite bipartite separated graph, following a paper by Exel and the author. In addition, we will review various results concerning the structure of the type semigroup for different dynamical systems.

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arXiv2025

The Leavitt inverse semigroup of a separated graph

We introduce and study a new inverse semigroup associated to a separated graph $(E,C)$, which we call the \emph{Leavitt inverse semigroup}. This semigroup is obtained as a quotient of the separated graph inverse semigroup $\mathcal{S}(E,C)$, introduced in our previous paper [9], and it provides a canonical inverse semigroup model for the tame Leavitt path algebra $\mathcal{L}_K^\mathrm{ab}(E,C)$ over a commutative unital ring $K$. Our first main result describes the Leavitt inverse semigroup $\mathcal{LI}(E,C)$ as a restricted semidirect product of the free group on the edges of $E$ acting partially on a certain semilattice, which is isomorphic to the semilattice of idempotents of $\mathcal{LI}(E,C)$. This description, given in terms of Leavitt--Munn trees, yields a normal form for the elements of $\mathcal{LI}(E,C)$. We obtain a normal form for elements of $\mathcal{L}_K^\mathrm{ab}(E,C)$, leading to explicit linear bases for $\mathcal{L}_K^\mathrm{ab}(E,C)$. Building on this and on the structural properties of $\mathcal{LI} (E,C)$, we prove that the natural homomorphism from $\mathcal{LI}(E,C)$ to $\mathcal{L}_K^\mathrm{ab}(E,C)$ is injective, so that $\mathcal{LI}(E,C)$ embeds as the inverse semigroup generated by the canonical partial isometries in $\mathcal{L}_K^\mathrm{ab}(E,C)$. Further applications include the determination of natural bases of the kernel $\mathcal Q$ of the natural map from the tame Cohn algebra $\mathcal{C}_K^\mathrm{ab} (E,C)$ to the tame Leavitt path algebtra $\mathcal{L}_K^\mathrm{ab} (E,C)$, the computation of the socle, and a characterization of the isolated points of the spectrum. Several examples, such as the Cuntz separated graph and free separations, are discussed to illustrate the theory.

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arXiv2024

Ideals, quotients, and continuity of the Cuntz semigroup for rings

In this paper we explore which part of the ideal lattice of a general ring is parametrized by its Cuntz semigroup $\mathrm{S}(R)$ and its ambient semigroup $\Lambda(R)$. We identify these classes of ideals as the quasipure ideals (a generalization of pure ideals) in the case of $\mathrm{S}(R)$, and what we term decomposable ideals in the case of $\Lambda(R)$. For an ($s$-)unital ring $R$, the latter class exhausts all ideals of the ring. We prove that these constructions behave well with respect to quotients. In order to study the passage to inductive limits, we introduce the classes of dense and left normal rings. We show that $\mathrm{S}(R)$ is an abstract Cu-semigroup whenever $R$ is left normal and, for such rings, the assignment $R\mapsto \mathrm{S}(R)$ is continuous. We prove a parallel result for $\Lambda(R)$ whenever $R$ is a dense ring.

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arXiv2024

Levels of cancellation for monoids and modules

Levels of cancellativity in commutative monoids $M$, determined by stable rank values in $\mathbb{Z}_{> 0} \cup \{\infty\}$ for elements of $M$, are investigated. The behavior of the stable ranks of multiples $ka$, for $k \in \mathbb{Z}_{> 0}$ and $a \in M$, is determined. In the case of a refinement monoid $M$, the possible stable rank values in archimedean components of $M$ are pinned down. Finally, stable rank in monoids built from isomorphism or other equivalence classes of modules over a ring is discussed.

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arXiv2024

Regular Ring Properties Degraded Through Inverse Limits

We give a number of constructions where inverse limits seriously degrade properties of regular rings, such as unit-regularity, diagonalisation of matrices, and finite stable rank. This raises the possibility of using inverse limits to answer the long standing Separativity Problem (in the negative).

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arXiv2024

The separativity problem in terms of varieties and diagonal reduction

We provide two new formulations of the separativity problem. First, it is known that separativity (and strong separativity) in von Neumann regular (and exchange) rings is tightly connected to unit-regularity of certain kinds of elements. By refining this information, we characterize separative regular rings in terms of a special type of inner inverse operation, which is defined via a single identity. This shows that the separative regular rings form a subvariety of the regular rings. The separativity problem reduces to the question of whether every element of the form $(1-aa')bac(1-a'a)$ in a regular ring is unit-regular, where $a'$ is an inner inverse for $a$. Second, it is known that separativity in exchange rings is equivalent to regular matrices over corner rings being reducible to diagonal matrices via elementary row and column operations. We show that for $2\times 2$ invertible matrices, four elementary operations are sufficient, and in general also necessary. Dropping the invertibility hypothesis, but specializing to separative regular rings, we show that three elementary row operations together with three elementary column operations are sufficient, and again in general also necessary. The separativity problem can subsequently be reframed in terms of the explicit number of operations needed to diagonally reduce.

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arXiv2024

Inverse semigroups of separated graphs and associated algebras

In this paper we introduce an inverse semigroup $\mathcal{S}(E,C)$ associated to a separated graph $(E,C)$ and describe its internal structure. In particular we show that it is strongly $E^*$-unitary and can be realized as a partial semidirect product of the form $\mathcal{Y}\rtimes\mathbb{F}$ for a certain partial action of the free group $\mathbb{F}=\mathbb{F}(E^1)$ on the edges of $E$ on a semilattice $\mathcal{Y}$ realizing the idempotents of $\mathcal{S}(E,C)$. In addition we also describe the spectrum as well as the tight spectrum of $\mathcal{Y}$. We then use the inverse semigroup $\mathcal{S}(E,C)$ to describe several "tame" algebras associated to $(E,C)$, including its Cohn algebra, its Leavitt-path algebra, and analogues in the realm of $C^*$-algebras, like the tame $C^*$-algebra $\mathcal{O}(E,C)$ and its Toeplitz extension $\mathcal{T}(E,C)$, proving that these algebras are canonically isomorphic to certain algebras attached to $\mathcal{S}(E,C)$. Our structural results on $\mathcal{S}(E,C)$ imply that these algebras can be realized as partial crossed products, revealing a great portion of their structure.

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arXiv2023

The Cuntz semigroup of a ring

For any ring $R$, we introduce an invariant in the form of a partially ordered abelian semigroup $\mathrm{S}(R)$ built from an equivalence relation on the class of countably generated projective modules. We call $\mathrm{S}(R)$ the Cuntz semigroup of the ring $R$. This construction is akin to the manufacture of the Cuntz semigroup of a C*-algebra using countably generated Hilbert modules. To circumvent the lack of a topology in a general ring $R$, we deepen our understanding of countably projective modules over $R$, thus uncovering new features in their direct limit decompositions, which in turn yields two equivalent descriptions of $\mathrm{S}(R)$. The Cuntz semigroup of $R$ is part of a new invariant $\mathrm{SCu}(R)$ which includes an ambient semigroup in the category of abstract Cuntz semigroups that provides additional information. We provide computations for both $\mathrm{S}(R)$ and $\mathrm{SCu}(R)$ in a number of interesting situations, such as unit-regular rings, semilocal rings, and in the context of nearly simple domains. We also relate our construcion to the Cuntz semigroup of a C*-algebra.

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arXiv2023

A correspondence between surjective local homeomorphisms and a family of separated graphs

We present a graph-theoretic model for dynamical systems $(X,\sigma)$ given by a surjective local homeomorphism $\sigma$ on a totally disconnected compact metrizable space $X$. In order to make the dynamics appear explicitly in the graph, we use two-colored Bratteli separated graphs as the graphs used to encode the information. In fact, our construction gives a bijective correspondence between such dynamical systems and a subclass of separated graphs which we call $l$-diagrams. This construction generalizes the well-known shifts of finite type, and leads naturally to the definition of a generalized finite shift. It turns out that any dynamical system $(X,\sigma)$ of our interest is the inverse limit of a sequence of generalized finite shifts. We also present a detailed study of the corresponding Steinberg and $C^*$ algebras associated with the dynamical system $(X,\sigma)$, and we use the above approximation of $(X,\sigma)$ to write these algebras as colimits of the associated algebras of the corresponding generalized finite shifts, which we call generalized finite shift algebras.

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arXiv2022

Free actions of groups on separated graph C*-algebras

In this paper we study free actions of groups on separated graphs and their \cstar{}algebras, generalizing previous results involving ordinary (directed) graphs. We prove a version of the Gross-Tucker Theorem for separated graphs yielding a characterization of free actions on separated graphs via a skew product of the (orbit) separated graph by a group labeling function. Moreover, we describe the C*-algebras associated to these skew products as crossed products by certain coactions coming from the labeling function on the graph. Our results deal with both the full and the reduced C*-algebras of separated graphs. To prove our main results we use several techniques that involve certain canonical conditional expectations defined on the C*-algebras of separated graphs and their structure as amalgamated free products of ordinary graph C*-algebras. Moreover, we describe Fell bundles associated with the coactions of the appearing labeling functions. As a byproduct of our results, we deduce that the \cstar{}algebras of separated graphs always have a canonical Fell bundle structure over the free group on their edges.

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arXiv2022

Leavitt path algebras of weighted and separated graphs

In this paper we show that Leavitt path algebras of weighted graphs and Leavitt path algebras of separated graphs are intimately related. We prove that any Leavitt path algebra $L(E,ω)$ of a row-finite vertex weighted graph $(E,ω)$ is $*$-isomorphic to the lower Leavitt path algebra of a certain bipartite separated graph $(E(ω),C(ω))$. For a general locally finite weighted graph $(E, ω)$, we show that a certain quotient $L_1(E,ω)$ of $L(E,ω)$ is $*$-isomorphic to an upper Leavitt path algebra of another bipartite separated graph $(E(w)_1,C(w)^1)$. We furthermore introduce the algebra $L^{\mathrm{ab}} (E,w)$, which is a universal tame $*$-algebra generated by a set of partial isometries. We draw some consequences of our results for the structure of ideals of $L(E,ω)$, and we study in detail two different maximal ideals of the Leavitt algebra $L(m,n)$.

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arXiv2021

The type semigroup, comparison and almost finiteness for ample groupoids

We prove that a minimal second countable ample groupoid has dynamical comparison if and only if its type semigroup is almost unperforated. Moreover, we investigate to what extent a not necessarily minimal almost finite groupoid has an almost unperforated type semigroup. Finally, we build a bridge between coarse geometry and topological dynamics by characterizing almost finiteness of the coarse groupoid in terms of a new coarsely invariant property for metric spaces, which might be of independent interest in coarse geometry. As a consequence, we are able to construct new examples of almost finite principal groupoids lacking other desirable properties, such as amenability or even a-T-menability. This behaviour is in stark contrast to the case of principal transformation groupoids associated to group actions.

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arXiv2020

L^2-Betti numbers arising from the lamplighter group

We apply a construction developed in a previous paper by the authors in order to obtain a formula which enables us to compute $\ell^2$-Betti numbers coming from a family of group algebras representable as crossed product algebras. As an application, we obtain a whole family of irrational $\ell^2$-Betti numbers arising from the lamplighter group algebra $K[\mathbb{Z}_2 \wr \mathbb{Z}]$, being $K$ a subfield of the complex numbers closed under complex conjugation. This procedure is constructive, in the sense that one has an explicit description of the elements realizing such irrational numbers. This extends the work made by Grabowski, who first computed irrational $\ell^2$-Betti numbers from the algebras $\mathbb{Q}[\mathbb{Z}_n \wr \mathbb{Z}]$, where $n \geq 2$ is a natural number. We also apply the techniques developed to the (generalized) odometer algebra $\mathcal{O}(\overline{n})$, where $\overline{n}$ is a supernatural number. We compute its $*$-regular closure, and this allows us to fully characterize the set of $\ell^2$-Betti numbers arising from $\mathcal{O}(\overline{n})$.

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arXiv2020

Approximating the group algebra of the lamplighter by infinite matrix products

In this paper, we introduce a new technique in the study of the $*$-regular closure of some specific group algebras $KG$ inside $\mathcal{U}(G)$, the $*$-algebra of unbounded operators affiliated to the group von Neumann algebra $\mathcal{N}(G)$. The main tool we use for this study is a general approximation result for a class of crossed product algebras of the form $C_K(X) \rtimes_T \mathbb{Z}$, where $X$ is a totally disconnected compact metrizable space, $T$ is a homeomorphism of $X$, and $C_K(X)$ stands for the algebra of locally constant functions on $X$ with values on an arbitrary field $K$. The connection between this class of algebras and a suitable class of group algebras is provided by Fourier transform. Utilizing this machinery, we study an explicit approximation for the lamplighter group algebra. This is used in another paper by the authors to obtain a whole family of $\ell^2$-Betti numbers arising from the lamplighter group, most of them transcendental.

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arXiv2020

Graded $K$-Theory, Filtered $K$-theory and the classification of graph algebras

We prove that an isomorphism of graded Grothendieck groups $K^{gr}_0$ of two Leavitt path algebras induces an isomorphism of a certain quotient of algebraic filtered $K$-theory and consequently an isomorphism of filtered $K$-theory of their associated graph $C^*$-algebras. As an application, we show that, since for a finite graph $E$ with no sinks, $K^{gr}_0\big(L(E)\big)$ of the Leavitt path algebra $L(E)$ coincides with Krieger's dimension group of its adjacency matrix $A_E$, our result relates the shift equivalence of graphs to the filtered $K$-theory and consequently gives that two arbitrary shift equivalent matrices give stably isomorphic graph $C^*$-algebras. This result was only known for irreducible graphs.

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arXiv2020

The Realization Problem for Finitely Generated Refinement Monoids

We show that every finitely generated conical refinement monoid can be represented as the monoid $\mathcal V(R)$ of isomorphism classes of finitely generated projective modules over a von Neumann regular ring $R$. To this end, we use the representation of these monoids provided by adaptable separated graphs. Given an adaptable separated graph $(E, C)$ and a field $K$, we build a von Neumann regular $K$-algebra $Q_K (E, C)$ and show that there is a natural isomorphism between the separated graph monoid $M(E, C)$ and the monoid $\mathcal V(Q_K (E, C))$.

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