arXiv subjects

Luiz Gustavo Cordeiro

Publications and source records attributed to Luiz Gustavo Cordeiro.

6 recordsLinked to original sources

arXiv2025

The uniform dimension of a monoid with applications to graph algebras

We adapt Goldie's concept of uniform dimensions from module theory over rings to $Γ$-monoids. A $Γ$-monoid $M$ is said to have uniform dimension $n$ if $n$ is the largest number of pairwise incomparable nonzero $Γ$-order ideals contained in $M$. Specializing to the talented monoid of a graph, we show that the uniform dimension provides a rough measure of how the graph branches out. Since for any order ideal $I$, its orthogonal ideal $I^\perp$ is the largest ideal incomparable to $I$, we study the notions of orthogonality and regularity, particularly when $I^{\perp\perp}=I$. We show that the freeness of the action of $\mathbb Z$ on the talented monoid of a graph is preserved under quotienting by a regular ideal. Furthermore, we determine the underlying hereditary and saturated sets that generate these ideals. These results unify recent studies on regular ideals of the corresponding Leavitt path algebras and graph $C^*$-algebras. We conclude that for graphs $E$ and $F$, if there is a $\mathbb Z$-monoid isomorphism $T_E\cong T_F$, then there is a one-to-one correspondence between the regular ideals of the associated Leavitt path algebras $L_K(E)$ and $L_K(F)$ (and similarly, $C^*(E)$ and $C^*(F)$). Since the talented monoid $T_E$ is the positive cone of the graded Grothendieck group $K_0^{gr}(L_K(E))$, this provides further evidence supporting the Graded Classification Conjecture.

math.RA↗

arXiv2021

The talented monoid of a directed graph with applications to graph algebras

It is a conjecture that for the class of Leavitt path algebras associated to finite directed graphs, their graded Grothendieck groups $K_0^{\mathrm{gr}}$ are a complete invariant. For a Leavitt path algebra $L_{\mathsf k}(E)$, with coefficient in a field ${\mathsf k}$, the monoid of the positive cone of $K_0^{\mathrm{gr}}(L_{\mathsf k}(E))$ can be described completely in terms of the graph $E$. In this note we further investigate the structure of this "talented monoid", showing how it captures intrinsic properties of the graph and hence the structure of its associated Leavitt path algebras. In particular, for the class of strongly connected graphs, we show that the notion of the period of a graph can be completely described via the talented monoid. As an application, we will give a finer characterisation of the purely infinite simple Leavitt path algebras in terms of properties of the associated graph. We show that graded isomorphism of algebras preserve the period of the graphs, and obtain results giving more evidence to the graded classification conjecture.

math.RA↗

arXiv2020

A general Banach-Stone type theorem and applications

One important class of tools in the study of the connections between algebraic and topological structures are the "Banach-Stone type theorems", which describe algebraic isomorphisms of algebras (or groups, lattices, etc.) of functions in terms of homeomorphisms between the underlying topological spaces. Several such theorems have been proven throughout the last century, however not all of them are comparable, and in particular no single one is the strongest. In this article, we describe a general framework which encompasses several of these results, and which allows for new applications related to groupoid algebras, and to groups of circle-valued functions. This is attained by a detailed study of "disjointness relations" on sets of functions, which play a central role (even if not explicitly) in previously-proven Banach-Stone type theorems.

math.GN↗

arXiv2019

Sectional algebras of semigroupoid bundles

In this article we use semigroupoids to describe a notion of algebraic bundles, mostly motivated by Fell ($C^*$-algebraic) bundles, and the sectional algebras associated to them. As the main motivational example, Steinberg algebras may be regarded as the sectional algebras of trivial (direct product) bundles. Several theorems which relate geometric and algebraic constructions - via the construction of a sectional algebra - are widely generalized: Direct products bundles by semigroupoids correspond to tensor products of algebras; Semidirect products of bundles correspond to "naïve" crossed products of algebras; Skew products of graded bundles correspond to smash products of graded algebras; Quotient bundles correspond to quotient algebras. Moreover, most of the results hold in the non-Hausdorff setting. In the course of this work, we generalize the definition of smash products to groupoid graded algebras. As an application, we prove that whenever $θ$ is a $\land$-preaction of a discrete inverse semigroupoid $S$ on an ample (possibly non-Hausdorff) groupoid $\mathcal{G}$, the Steinberg algebra of the associated groupoid of germs is naturally isomorphic to a crossed product of the Steinberg algebra of $\mathcal{G}$ by $S$. This is a far-reaching generalization of analogous results which had been proven in particular cases.

math.RA↗

arXiv2019

Étale inverse semigroupoids - the fundamentals

In this article we will study semigroupoids, and more specifically inverse semigroupoids. These are a common generalization to both inverse semigroups and groupoids, and provide a natural language on which several types of dynamical structures may be described. Moreover, this theory allows us to precisely compare and simultaneously generalize aspects of both the theories of inverse semigroups and groupoids. We begin by comparing and settling the differences between two notions of semigroupoids which appear in the literature (one by Tilson and another by Exel). We specialize this study to inverse semigroupoids, and in particular an analogue of the Vagner-Preston Theorem is obtained. This representation theorem leads to natural notions of actions, and more generally $\land$-preactions and partial actions, of étale inverse semigroupoids, which generalize topological actions of inverse semigroups and continuous actions of étale groupoids. Many constructions which are commonplace in the theories of inverse semigroups and groupoids are also generalized, and their categorical properties made explicit. We finish this paper with a version of non-commutative Stone duality for ample inverse semigroupoids, which utilizes several of the aforementioned constructions.

math.DS↗

arXiv2018

The dynamics of partial inverse semigroup actions

Given an inverse semigroup $S$ endowed with a partial action on a topological space $X$, we construct a groupoid of germs $S\ltimes X$ in a manner similar to Exel's groupoid of germs, and similarly a partial action of $S$ on an algebra $A$ induces a crossed product $A\rtimes S$. We then prove, in the setting of partial actions, that if $X$ is locally compact Hausdorff and zero-dimensional, then the Steinberg algebra of the groupoid of germs $S\ltimes X$ is isomorphic to the crossed product $A_R(X)\rtimes S$, where $A_R(X)$ is the Steinberg algebra of $X$. We also prove that the converse holds, that is, that under natural hypotheses, crossed products of the form $A_R(X)\rtimes S$ are Steinberg algebras of appropriate groupoids of germs of the form $S\ltimes X$. We introduce a new notion of topologically principal partial actions, which correspond to topologically principal groupoids of germs, and study orbit equivalence for these actions in terms of isomorphisms of the corresponding groupoids of germs. This generalizes previous work of the first-named author as well as from others, which dealt mostly with global actions of semigroups or partial actions of groups. We finish the article by comparing our notion of orbit equivalence of actions and orbit equivalence of graphs.

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