Related papers: Leavitt path algebras having Unbounded Generating …
We introduce a graded homology theory for graded \'etale groupoids. For $\mathbb Z$-graded groupoids, we establish an exact sequence relating the graded zeroth-homology to non-graded one. Specialising to the arbitrary graph groupoids, we…
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…
Leavitt path algebras associate to directed graphs a $\mathbb Z$-graded algebra and in their simplest form recover the Leavitt algebras $L(1,k)$. In this note, we first study this $\mathbb Z$-grading and characterize the ($\mathbb…
This survey reports on current progress of programs to classify graph C*-algebras and Leavitt path algebras up to Morita equivalence using K-theory. Beginning with an overview and some history, we trace the development of the classification…
If $K$ is a field with involution and $E$ an arbitrary graph, the involution from $K$ naturally induces an involution of the Leavitt path algebra $L_K(E).$ We show that the involution on $L_K(E)$ is proper if the involution on $K$ is…
We introduce a notion of ideal-related K-theory for rings, and use it to prove that if two complex Leavitt path algebras are Morita equivalent (respectively, isomorphic), then the ideal-related K-theories (respectively, the unital…
We determine the Gelfand-Kirillov dimension of a weighted Leavitt path algebra $L_K(E,w)$ where $K$ is a field and $(E,w)$ a finite weighted graph. Further we show that a finite-dimensional weighted Leavitt path algebra over a field $K$ is…
In this paper we address the classification problem for purely infinite simple Leavitt path algebras of finite graphs over a field $\ell$. Each graph $E$ has associated a Leavitt path $\ell$-algebra $L(E)$. There is an open question which…
Given a finite group $G$, the invariably generating graph of $G$ is defined as the undirected graph in which the vertices are the nontrivial conjugacy classes of $G$, and two classes are adjacent if and only if they invariably generate $G$.…
Given a simple vertex algebra A and a reductive group G of automorphisms of A, the invariant subalgebra A^G is strongly finitely generated in most examples where its structure is known. This phenomenon is subtle, and is generally not true…
We prove that the Bowen-Franks group classifies the Leavitt path algebras of purely infinite simple finite graphs over a regular supercoherent commutative ring with involution where $2$ is invertible, equipped with their standard…
When the theory of Leavitt path algebras was already quite advanced, it was discovered that some of the more difficult questions were susceptible to a new approach using topological groupoids. The main result that makes this possible is…
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,…
We show that the graph construction used to prove that a gauge-invariant ideal of a graph C*-algebra is isomorphic to a graph C*-algebra, and also used to prove that a graded ideal of a Leavitt path algebra is isomorphic to a Leavitt path…
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…
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…
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…
In this paper we study the graded version of Naimark's problem for Leavitt path algebras considering them as $\mathbb{Z}$-graded algebras. Several characterizations are obtained of a Leavitt path algebra $L$ of an arbitrary graph $E$ over a…
We describe how boundary paths in a graph can be used to construct irreducible representations of the associated graph C*-algebra and the associated Leavitt path algebra. We use this construction to establish two sets of results: First, we…
In this work, we propose to study noncommutative geometry using the language of categories of sheaves of algebras with polynomial identities and their properties, introducing new (graded) noncommutative geometries. These include, for…