Related papers: *-Regular Leavitt path algebras of arbitrary graph…

Leavitt path algebras L of an arbitrary graph E over a field K satisfying a polynomial identity are completely characterized both in graph-theoretic and algebraic terms. When E is a finite graph, L satisfying a polynomial identity is shown…

In this paper, we give a complete characterization of Leavitt path algebras which are graded $\Sigma $-$V$ rings, that is, rings over which a direct sum of arbitrary copies of any graded simple module is graded injective. Specifically, we…

For any field $K$ and for a completely arbitrary graph $E$, we characterize the Leavitt path algebras $L_K(E)$ that are indecomposable (as a direct sum of two-sided ideals) in terms of the underlying graph. When the algebra decomposes, it…

Let $R$ be a unital ring, let $E$ be a directed graph and recall that the Leavitt path algebra $L_R(E)$ carries a natural $\mathbb{Z}$-gradation. We show that $L_R(E)$ is strongly $\mathbb{Z}$-graded if and only if $E$ is row-finite, has no…

Let $E$ be a finite directed graph, and let $I$ be the poset obtained as the antisymmetrization of its set of vertices with respect to a pre-order $\le$ that satisfies $v\le w$ whenever there exists a directed path from $w$ to $v$. Assuming…

In sharp contrast to the Abrams-Rangaswamy Theorem that the only von Neumann regular Leavitt path algebras are exactly those associated to acyclic graphs, here we prove that the Leavitt path algebra of any arbitrary graph is a graded von…

We prove that if E and F are graphs with a finite number of vertices and an infinite number of edges, if K is a field, and if L_K(E) and L_K(F) are simple Leavitt path algebras, then L_K(E) is Morita equivalent to L_K(F) if and only if…

We prove Leavitt path algebra versions of the two uniqueness theorems of graph C*-algebras. We use these uniqueness theorems to analyze the ideal structure of Leavitt path algebras and give necessary and sufficient conditions for their…

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…

Let $E$ be a row-finite quiver and let $E_0$ be the set of vertices of $E$; consider the adjacency matrix $N'_E=(n_{ij})\in\Z^{(E_0\times E_0)}$, $n_{ij}=#\{$ arrows from $i$ to $j\}$. Write $N^t_E$ and 1 for the matrices $\in…

Let $K$ be a field. We characterise the row-finite weighted graphs $(E,w)$ such that the weighted Leavitt path algebra $L_K(E,w)$ is isomorphic to an unweighted Leavitt path algebra. Moreover, we prove that if $L_K(E,w)$ is locally finite,…

Let $E$ be a directed graph, $K$ any field, and let $L_K(E)$ denote the Leavitt path algebra of $E$ with coefficients in $K$. For each rational infinite path $c^\infty$ of $E$ we explicitly construct a projective resolution of the…

We prove an algebraic version of the Gauge-Invariant Uniqueness Theorem, a result which gives information about the injectivity of certain homomorphisms between ${\mathbb Z}$-graded algebras. As our main application of this theorem, we…

In this paper we completely describe graphically Leavitt path algebras with bounded index of nilpotence. We show that the Leavitt path algebra $L_{K}(E)$ has index of nilpotence at most $n$ if and only if no cycle in the graph $E$ has an…

Any finite dimensional semisimple algebra A over a field K is isomorphic to a direct sum of finite dimensional full matrix rings over suitable division rings. In this paper we will consider the special case where all division rings are…

In this article, we give necessary and sufficient conditions under which the Leavitt path algebra $L_K(\mathcal{G})$ of an ultragraph $\mathcal{G}$ over a field $K$ is purely infinite simple and that it is von Neumann regular. Consequently,…

Suppose that $R$ is an associative unital ring and that $E=(E^0,E^1,r,s)$ is a directed graph. Utilizing results from graded ring theory we show, that the associated Leavitt path algebra $L_R(E)$ is simple if and only if $R$ is simple,…

We characterize Leavitt path algebras which are exchange rings in terms of intrinsic properties of the graph and show that the values of the stable rank for these algebras are 1, 2 or $\infty$. Concrete criteria in terms of properties of…

Let $G$ be a group and $\ell$ a commutative unital $\ast$-ring with an element $\lambda \in \ell$ such that $\lambda + \lambda^\ast = 1$. We introduce variants of hermitian bivariant $K$-theory for $\ast$-algebras equipped with a $G$-action…

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…