arXiv subjects

Lia Vas

Publications and source records attributed to Lia Vas.

At least 19 recordsLinked to original sources

arXiv2026

The Graded Classification Conjecture holds for graphs with disjoint cycles

The Graded Classification Conjecture (GCC) states that the pointed $K_0^{\operatorname{gr}}$-group is a complete invariant of the Leavitt path algebras of finite graphs when these algebras are considered with their natural grading by $\mathbb Z.$ The conjecture has previously been shown to hold in some special cases. The main result of the paper shows that the GCC holds for a significantly more general class of graphs included in the class of graphs with disjoint cycles. In particular, our result holds for finite graphs with disjoint cycles. We show the main result also for graph $C^*$-algebras. As a consequence, the graded version of the Isomorphism Conjecture holds for the class of graphs we consider. Besides showing the conjecture for the class of graphs we consider, we realize the Grothendieck $\mathbb Z$-group isomorphism by a specific graded $*$-isomorphism. In particular, we introduce a series of graph operations which preserve the graded $*$-isomorphism class of their algebras. After performing these operations on a graph, we obtain well-behaved ``representative'' graphs, which we call canonical forms. We define an equivalence $\approx$ on graphs such that $E\approx F$ holds when there are isomorphic canonical forms of $E$ and $F$ and we show that the condition $E\approx F$ is equivalent to the existence of an isomorphism $f$ of the Grothendieck $\mathbb Z$-groups of the algebras of $E$ and $F$ in the appropriate category. As $E\approx F$ can be realized by a finite series of specific graph operations, any such isomorphism $f$ can be realized by an explicit graded $*$-algebra isomorphism. Thus, we describe the graded ($*$-)isomorphism classes of the algebras of graphs we consider. Besides the ties to symbolic dynamics and Williams' Problem, such a description is relevant for the active program of classification of graph $C^*$-algebras.

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arXiv2023

Porcupine-quotient graphs, the fourth primary color, and graded composition series of Leavitt path algebras

If $E$ is a directed graph, $K$ is a field, and $I$ is a graded ideal of the Leavitt path algebra $L_K(E),$ $I$ is completely determined by an admissible pair $(H,S)$ of two sets of vertices of $E$. The ideal $I=I(H,S)$ is graded isomorphic to the Leavitt path algebra of the {\em porcupine graph} of $(H,S)$ and the quotient $L_K(E)/I$ is graded isomorphic to the Leavitt path algebra of the {\em quotient graph} of $(H,S).$ We present a construction which generalizes both constructions and enables one to consider quotients of graded ideals: if $(H,S)$ and $(G,T)$ are admissible pairs such that $I(H,S)\subseteq I(G,T)$, we define the {\em porcupine-quotient graph} $(G,T)/(H,S)$ such that its Leavitt path algebra is graded isomorphic to the quotient $I(G,T)/I(H,S).$ Using the porcupine-quotient construction, the existence of a graded composition series of $L_K(E)$ is equivalent to the existence of a finite chain of admissible pairs of $E,$ starting with the trivial and ending with the improper pair, such that the quotient of two consecutive pairs is cofinal (a graph is cofinal exactly when its Leavitt path algebra is graded simple). We characterize the existence of such a composition series with a set of conditions which also provides an algorithm for obtaining such a series. The conditions are presented in terms of four types of vertices which are all ``terminal'' in a certain sense. Three types are often referred to as the three primary colors and the fourth type is new. As a corollary, a unital Leavitt path algebra has a graded composition series. We show that the existence of a composition series of $E$ is equivalent to the existence of a composition series of the graph monoid $M_E$ as well as a composition series of the talented monoid $M_E^Γ.$ An ideal of $M_E^Γ$ is minimal exactly when it is generated by the element of $M_E^Γ$ corresponding to a terminal vertex.

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arXiv2023

Graph characterization of the annihilator ideals of Leavitt path algebras

If $E$ is a graph and $K$ is a field, we consider an ideal $I$ of the Leavitt path algebra $L_K(E)$ of $E$ over $K$. We describe the admissible pair corresponding to the smallest graded ideal which contains $I$ where the grading in question is the natural grading of $L_K(E)$ by $\mathbb Z$. Using this description, we show that the right and the left annihilators of $I$ are equal (which can be somewhat surprising given that $I$ may not be self-adjoint). In particular, we establish that both annihilators correspond to the same admissible pair and its description produces the characterization from the title. Then, we turn to the property that the right (equivalently left) annihilator of any ideal is a direct summand and recall that a unital ring with this property is said to be quasi-Baer. We exhibit a condition on $E$ which is equivalent to unital $L_K(E)$ having this property.

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arXiv2023

Baer and Baer *-ring characterizations of Leavitt path algebras

We characterize Leavitt path algebras which are Rickart, Baer, and Baer $*$-rings in terms of the properties of the underlying graph. In order to treat non-unital Leavitt path algebras as well, we generalize these annihilator-related properties to locally unital rings and provide a more general characterizations of Leavitt path algebras which are locally Rickart, locally Baer, and locally Baer $*$-rings. Leavitt path algebras are also graded rings and we formulate the graded versions of these annihilator-related properties and characterize Leavitt path algebras having those properties as well. Our characterizations provide a quick way to generate a wide variety of examples of rings. For example, creating a Baer and not a Baer $*$-ring, a Rickart $*$-ring which is not Baer, or a Baer and not a Rickart $*$-ring, is straightforward using the graph-theoretic properties from our results. In addition, our characterizations showcase more properties which distinguish behavior of Leavitt path algebras from their $C^*$-algebra counterparts. For example, while a graph $C^*$-algebra is Baer (and a Baer $*$-ring) if and only if the underlying graph is finite and acyclic, a Leavitt path algebra is Baer if and only if the graph is finite and no cycle has an exit, and it is a Baer $*$-ring if and only if the graph is a finite disjoint union of graphs which are finite and acyclic or loops.

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arXiv2023

Realization of graded matrix algebras as Leavitt path algebras

While every matrix algebra over a field $K$ can be realized as a Leavitt path algebra, this is not the case for every graded matrix algebra over a graded field. We provide a complete description of graded matrix algebras over a field, trivially graded by the ring of integers, which are graded isomorphic to Leavitt path algebras. As a consequence, we show that there are graded corners of Leavitt path algebras which are not graded isomorphic to Leavitt path algebras. This contrasts a recent result stating that every corner of a Leavitt path algebra of a finite graph is isomorphic to a Leavitt path algebra. If $R$ is a finite direct sum of graded matricial algebras over a trivially graded field and over naturally graded fields of Laurent polynomials, we also present conditions under which $R$ can be realized as a Leavitt path algebra.

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arXiv2023

Comparability in the graph monoid

Let $Γ$ be the infinite cyclic group on a generator $x.$ To avoid confusion when working with $\mathbb Z$-modules which also have an additional $\mathbb Z$-action, we consider the $\mathbb Z$-action to be a $Γ$-action instead. Starting from a directed graph $E$, one can define a cancellative commutative monoid $M_E^Γ$ with a $Γ$-action which agrees with the monoid structure and a natural order. The order and the action enable one to label each nonzero element as being exactly one of the following: comparable (periodic or aperiodic) or incomparable. We comprehensively pair up these element features with the graph-theoretic properties of the generators of the element. We also characterize graphs such that every element of $M_E^Γ$ is comparable, periodic, graphs such that every nonzero element of $M_E^Γ$ is aperiodic, incomparable, graphs such that no nonzero element of $M_E^Γ$ is periodic, and graphs such that no element of $M_E^Γ$ is aperiodic. The Graded Classification Conjecture can be formulated to state that $M_E^Γ$ is a complete invariant of the Leavitt path algebra $L_K(E)$ of $E$ over a field $K.$ Our characterizations indicate that the Graded Classification Conjecture may have a positive answer since the properties of $E$ are well reflected by the structure of $M_E^Γ.$ Our work also implies that some results of [R. Hazrat, H. Li, The talented monoid of a Leavitt path algebra, J. Algebra, 547 (2020) 430-455] hold without requiring the graph to be row-finite.

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arXiv2022

Every graded ideal of a Leavitt path algebra is graded isomorphic to a Leavitt path algebra

We show that every graded ideal of a Leavitt path algebra is graded isomorphic to a Leavitt path algebra. It is known that a graded ideal $I$ of a Leavitt path algebra is isomorphic to the Leavitt path algebra of a graph, known as the generalized hedgehog graph, which is defined based on certain sets of vertices uniquely determined by $I$. However, this isomorphism may not be graded. We show that replacing the short "spines" of the generalized hedgehog graph with possibly fewer, but then necessarily longer spines, we obtain a graph (which we call the porcupine graph) such that its Leavitt path algebra is graded isomorphic to $I$. Our proof adapts to show that for every closed gauge-invariant ideal $J$ of a graph $C^*$-algebra, there is a gauge-invariant $*$-isomorphism mapping the graph $C^*$-algebra of the porcupine graph of $J$ onto $J.$

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arXiv2022

The functor $K_0^{\operatorname{gr}}$ is full and only weakly faithful

The Graded Classification Conjecture states that the pointed $K_0^{\operatorname{gr}}$-group is a complete invariant of the Leavitt path algebras of finite graphs when these algebras are considered with their natural grading by $\mathbb Z.$ The strong version of this conjecture states that the functor $K_0^{\operatorname{gr}}$ is full and faithful when considered on the category of Leavitt path algebras of finite graphs and their graded homomorphisms modulo conjugations by invertible elements of the zero components. We show that the functor $K_0^{\operatorname{gr}}$ is full for the unital Leavitt path algebras of countable graphs and that it is faithful (modulo specified conjugations) only in a certain weaker sense.

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arXiv2022

Graded irreducible representations of Leavitt path algebras: a new type and complete classification

We present a new class of graded irreducible representations of a Leavitt path algebra. This class is new in the sense that its representation space is not isomorphic to any of the existing simple Chen modules. The corresponding graded simple modules complete the list of Chen modules which are graded, creating an exhaustive class: the annihilator of any graded simple module is equal to the annihilator of either a graded Chen module or a module of this new type. Our characterization of graded primitive ideals of a Leavitt path algebra in terms of the properties of the underlying graph is the main tool for proving the completeness of such classification. We also point out a problem with the characterization of primitive ideals of a Leavitt path algebra in [K. M. Rangaswamy, Theory of prime ideals of Leavitt path algebras over arbitrary graphs, J. Algebra 375 (2013), 73 -- 90].

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arXiv2022

Annihilator ideals of graph algebras

If $I$ is a (two-sided) ideal of a ring $R$, we let $\operatorname{ann}_l(I)=\{r\in R\mid rI=0\},$ $\operatorname{ann}_r(I)=\{r\in R\mid Ir=0\},$ and $\operatorname{ann}(I)=\operatorname{ann}_l(I)\cap \operatorname{ann}_r(I)$ be the left, the right and the double annihilators. An ideal $I$ is said to be an annihilator ideal if $I=\operatorname{ann}(J)$ for some ideal $J$ (equivalently, $\operatorname{ann}(\operatorname{ann}(I))=I$). We study annihilator ideals of Leavitt path algebras and graph $C^*$-algebras. Let $L_K(E)$ be the Leavitt path algebra of a graph $E$ over a field $K.$ If $I$ is an ideal of $L_K(E),$ it has recently been shown that $\operatorname{ann}(I)$ is a graded ideal (with respect to the natural grading of $L_K(E)$ by $\mathbb Z$). We note that $\operatorname{ann}_l(I)$ and $\operatorname{ann}_r(I)$ are also graded. For a graded ideal $I,$ we describe $\operatorname{ann}(I)$ in terms of the properties of a pair of sets of vertices of $E,$ known as an admissible pair, which naturally corresponds to $I.$ Using such a description, we present properties of $E$ which are equivalent with the requirement that each graded ideal of $L_K(E)$ is an annihilator ideal. We show that the same properties of $E$ are also equivalent with each of the following conditions: (1) The lattice of graded ideals of $L_K(E)$ is a Boolean algebra; (2) Each closed gauge-invariant ideal of $C^*(E)$ is an annihilator ideal; (3) The lattice of closed gauge-invariant ideals of $C^*(E)$ is a Boolean algebra. In addition, we present properties of $E$ which are equivalent with each of the following conditions: (1) Each ideal of $L_K(E)$ is an annihilator ideal; (2) The lattice of ideals of $L_K(E)$ is a Boolean algebra; (3) Each closed ideal of $C^*(E)$ is an annihilator ideal; (4) The lattice of closed ideals of $C^*(E)$ is a Boolean algebra.

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arXiv2022

Graded cancellation properties of graded rings and graded unit-regular Leavitt path algebras

We raise the following general question regarding a ring graded by a group: "If $P$ is a ring-theoretic property, how does one define the graded version $P_{\operatorname{gr}}$ of the property $P$ in a meaningful way?". Some properties of rings have straightforward and unambiguous generalizations to their graded versions and these generalizations satisfy all the matching properties of the nongraded case. If $P$ is either being unit-regular, having stable range 1 or being directly finite, that is not the case. The first part of the paper addresses this issue. Searching for appropriate generalizations, we consider graded versions of cancellation, internal cancellation, substitution, and module-theoretic direct finiteness. In the second part of the paper, we turn to Leavitt path algebras. If $K$ is a trivially graded field and $E$ is an oriented graph, the Leavitt path algebra $L_K(E)$ is naturally graded by the ring of integers. If $E$ is a finite graph, we present a property of $E$ which is equivalent with $L_K(E)$ being graded unit-regular. This property critically depends on the lengths of paths to cycles making it stand out from other known graph conditions which characterize algebraic properties of $L_K(E).$ It also further illustrates that graded unit-regularity is quite restrictive in comparison to the alternative generalization of unit-regularity which we consider in the first part of the paper.

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arXiv2022

Crossed product Leavitt path algebras

If $E$ is a directed graph and $K$ is a field, the Leavitt path algebra $L_K(E)$ of $E$ over $K$ is naturally graded by the group of integers $\mathbb Z.$ We formulate properties of the graph $E$ which are equivalent with $L_K(E)$ being a crossed product, a skew group ring, or a group ring with respect to this natural grading. We state this main result so that the algebra properties of $L_K(E)$ are also characterized in terms of the pre-ordered group properties of the Grothendieck $\mathbb Z$-group of $L_K(E)$. If $E$ has finitely many vertices, we characterize when $L_K(E)$ is strongly graded in terms of the properties of $K_0^Γ(L_K(E)).$ Our proof also provides an alternative to the known proof of the equivalence $L_K(E)$ is strongly graded if and only if $E$ has no sinks for a finite graph $E.$ We also show that, if unital, the algebra $L_K(E)$ is strongly graded and graded unit-regular if and only if $L_K(E)$ is a crossed product. In the process of showing the main result, we obtain conditions on a group $Γ$ and a $Γ$-graded division ring $K$ equivalent with the requirements that a $Γ$-graded matrix ring $R$ over $K$ is strongly graded, a crossed product, a skew group ring, or a group ring. We characterize these properties also in terms of the action of the group $Γ$ on the Grothendieck $Γ$-group $K_0^Γ(R).$

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arXiv2022

Simplicial and dimension groups with group action and their realization

We define simplicial and dimension $Γ$-groups, the generalizations of simplicial and dimension groups to the case when these groups have an action of an arbitrary group $Γ.$ Assuming that the integral group ring of $Γ$ is Noetherian, we show that every dimension $Γ$-group is isomorphic to a direct limit of a directed system of simplicial $Γ$-groups and that the limit can be taken in the category of ordered groups with order-units or generating intervals. We adapt Hazrat's definition of the Grothendieck $Γ$-group $K_0^Γ(R)$ for a $Γ$-graded ring $R$ to the case when $Γ$ is not necessarily abelian. If $G$ is a pre-ordered abelian group with an action of $Γ$ which agrees with the pre-ordered structure, we say that $G$ is {\em realized} by a $Γ$-graded ring $R$ if $K_0^Γ(R)$ and $G$ are isomorphic as pre-ordered $Γ$-groups with an isomorphism which preserves order-units or generating intervals. We show that every simplicial $Γ$-group with an order-unit can be realized by a graded matricial ring over a $Γ$-graded division ring. If the integral group ring of $Γ$ is Noetherian, we realize a countable dimension $Γ$-group with an order-unit or a generating interval by a $Γ$-graded ultramatricial ring over a $Γ$-graded division ring. We also relate our results to graded rings with involution which give rise to Grothendieck $Γ$-groups with actions of both $Γ$ and $\mathbb Z_2$. We adapt the Realization Problem for von Neumann regular rings to graded rings and concepts from this work and discuss some other questions.

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arXiv2021

Cancellation properties of graded and nonunital rings. Graded clean and graded exchange Leavitt path algebras

Various authors have been generalizing some unital ring properties to nonunital rings. We consider properties related to cancellation of modules (being unit-regular, having stable range one, being directly finite, exchange, or clean) and their "local" versions. We explore their relationships and extend the defined concepts to graded rings. With graded clean and graded exchange rings suitably defined, we study how these properties behave under the formation of graded matrix rings. We exhibit properties of a graph $E$ which are equivalent to the unital Leavitt path algebra $L_K(E)$ being graded clean. We also exhibit some graph properties which are necessary and some which are sufficient for $L_K(E)$ to be graded exchange.

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arXiv2018

K-theory Classification of Graded Ultramatricial Algebras with Involution

We consider a generalization $K_0^{\operatorname{gr}}(R)$ of the standard Grothendieck group $K_0(R)$ of a graded ring $R$ with involution. If $Γ$ is an abelian group, we show that $K_0^{\operatorname{gr}}$ completely classifies graded ultramatricial $*$-algebras over a $Γ$-graded $*$-field $A$ such that (1) each nontrivial graded component of $A$ has a unitary element in which case we say that $A$ has enough unitaries, and (2) the zero-component $A_0$ is 2-proper (for any $a,b\in A_0,$ $aa^*+bb^*=0$ implies $a=b=0$) and $*$-pythagorean (for any $a,b\in A_0,$ $aa^*+bb^*=cc^*$ for some $c\in A_0$). If the involutive structure is not considered, our result implies that $K_0^{\operatorname{gr}}$ completely classifies graded ultramatricial algebras over any graded field $A.$ If the grading is trivial and the involutive structure is not considered, we obtain some well known results as corollaries. If $R$ and $S$ are graded matricial $*$-algebras over a $Γ$-graded $*$-field $A$ with enough unitaries and $f: K_0^{\operatorname{gr}}(R)\to K_0^{\operatorname{gr}}(S)$ is a contractive $\mathbb Z[Γ]$-module homomorphism, we present a specific formula for a graded $*$-homomorphism $ϕ: R\to S$ with $K_0^{\operatorname{gr}}(ϕ) = f.$ If the grading is trivial and the involutive structure is not considered, our constructive proof implies the known results with existential proofs. As an application of our results, we show that the graded version of the Isomorphism Conjecture holds for a class of Leavitt path algebras: if $E$ and $F$ are countable, row-finite, no-exit graphs in which every path ends in a sink or a cycle and $K$ is a 2-proper and $*$-pythagorean field, then the Leavitt path algebras $L_K(E)$ and $L_K(F)$ are isomorphic as graded rings if any only if they are isomorphic as graded $*$-algebras.

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arXiv2017

Graded chain conditions and Leavitt path algebras of no-exit graphs

We obtain a complete structural characterization of Cohn-Leavitt algebras over no-exit objects as graded involutive algebras. Corollaries of this result include graph-theoretic conditions characterizing when a Leavitt path algebra is a directed union of (graded) matricial algebras over the underlying field and over the algebra of Laurent polynomials and when the monoid of isomorphism classes of finitely generated projective modules is atomic and cancellative. We introduce the non-unital generalizations of graded analogues of noetherian and artinian rings, graded locally noetherian and graded locally artinian rings, and characterize graded locally noetherian and graded locally artinian Leavitt path algebras without any restriction on the cardinality of the graph. As a consequence, we relax the assumptions of the Abrams-Aranda-Perera-Siles characterization of locally noetherian and locally artinian Leavitt path algebras.

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arXiv2015

Canonical traces and directly finite Leavitt path algebras

Motivated by the study of traces on graph $C^*$-algebras, we consider traces (additive, central maps) on Leavitt path algebras, the algebraic counterparts of graph $C^*$-algebras. In particular, we consider traces which vanish on nonzero graded components of a Leavitt path algebra and refer to them as {\em canonical} since they are uniquely determined by their values on the vertices. A desirable property of a $\mathbb{C}$-valued trace on a $C^*$-algebra is that the trace of an element of the positive cone is nonnegative. We adapt this property to traces on a Leavitt path algebra $L_K(E)$ with values in any involutive ring. We refer to traces with this property as positive. If a positive trace is injective on positive elements, we say that it is faithful. We characterize when a canonical, $K$-linear trace is positive and when it is faithful in terms of its values on the vertices. As a consequence, we obtain a bijective correspondence between the set of faithful, gauge invariant, $\mathbb{C}$-valued (algebra) traces on $L_{\mathbb{C}}(E)$ of a countable graph $E$ and the set of faithful, semifinite, lower semicontinuous, gauge invariant (operator theory) traces on the corresponding graph $C^*$-algebra $C^*(E)$. With the direct finite condition (i.e $xy=1$ implies $yx=1$) for unital rings adapted to rings with local units, we characterize directly finite Leavitt path algebras as exactly those having the underlying graphs in which no cycle has an exit. Our proof involves consideration of "local" Cohn-Leavitt subalgebras of finite subgraphs. Lastly, we show that, while related, the class of locally noetherian, the class of directly finite, and the class of Leavitt path algebras which admit a faithful trace are different in general.

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arXiv2015

Traces on Semigroup Rings and Leavitt Path Algebras

The trace on matrix rings, along with the augmentation map and Kaplansky trace on group rings, are some of the many examples of linear functions on algebras that vanish on all commutators. We generalize and unify these examples by studying traces on (contracted) semigroup rings over commutative rings. We show that every such ring admits a minimal trace (i.e., one that vanishes only on sums of commutators), classify all minimal traces on these rings, and give applications to various classes of semigroup rings and quotients thereof. We then study traces on Leavitt path algebras (which are quotients of contracted semigroup rings), where we describe all linear traces in terms of central maps on graph inverse semigroups and, under mild assumptions, those Leavitt path algebras that admit faithful traces.

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