Carlsen [‘$\ast $-isomorphism of Leavitt path algebras over $\Bbb Z$’, Adv. Math. 324 (2018), 326–335] showed that any $\ast $-homomorphism between Leavitt path algebras over $\mathbb Z$ is automatically diagonal preserving and hence induces an isomorphism of boundary path groupoids. His result works over conjugation-closed subrings of $\mathbb C$ enjoying certain properties. In this paper, we characterise the rings considered by Carlsen as precisely those rings for which every $\ast $-homomorphism of algebras of Hausdorff ample groupoids is automatically diagonal preserving. Moreover, the more general groupoid result has a simpler proof.

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 may 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 characterisation 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.

Suppose that R is an associative unital ring and that $E=(E^0,E^1,r,s)$ is a directed graph. Using 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, $E^0$ has no nontrivial hereditary and saturated subset, and every cycle in E has an exit. We also give a complete description of the centre of a simple Leavitt path algebra.

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 $L(E,\omega )$ of a row-finite vertex weighted graph $(E,\omega )$ is $*$-isomorphic to the lower Leavitt path algebra of a certain bipartite separated graph $(E(\omega ),C(\omega ))$. For a general locally finite weighted graph $(E, \omega )$, we show that a certain quotient $L_1(E,\omega )$ of $L(E,\omega )$ 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 $L(E,\omega )$, and we study in detail two different maximal ideals of the Leavitt algebra $L(m,n)$.

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 generalised 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 generalised hedgehog graph with possibly fewer, but then necessarily longer spines, we obtain a graph (which we call the porcupine graph) whose Leavitt path algebra is graded isomorphic to I. Our proof can be adapted 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 $J.$

For a field K, let $\mathcal {R}$ denote the Jacobson algebra $K\langle X, Y \ | \ XY=1\rangle $ . We give an explicit construction of the injective envelope of each of the (infinitely many) simple left $\mathcal {R}$ -modules. Consequently, we obtain an explicit description of a minimal injective cogenerator for $\mathcal {R}$ . Our approach involves realizing $\mathcal {R}$ up to isomorphism as the Leavitt path K-algebra of an appropriate graph $\mathcal {T}$ , which thereby allows us to utilize important machinery developed for that class of algebras.

We show that a directed graph $E$ is a finite graph with no sinks if and only if, for each commutative unital ring $R$, the Leavitt path algebra $L_{R}(E)$ is isomorphic to an algebraic Cuntz–Krieger algebra if and only if the $C^{\ast }$-algebra $C^{\ast }(E)$ is unital and $\text{rank}(K_{0}(C^{\ast }(E)))=\text{rank}(K_{1}(C^{\ast }(E)))$. Let $k$ be a field and $k^{\times }$ be the group of units of $k$. When $\text{rank}(k^{\times })<\infty$, we show that the Leavitt path algebra $L_{k}(E)$ is isomorphic to an algebraic Cuntz–Krieger algebra if and only if $L_{k}(E)$ is unital and $\text{rank}(K_{1}(L_{k}(E)))=(\text{rank}(k^{\times })+1)\text{rank}(K_{0}(L_{k}(E)))$. We also show that any unital $k$-algebra which is Morita equivalent or stably isomorphic to an algebraic Cuntz–Krieger algebra, is isomorphic to an algebraic Cuntz–Krieger algebra. As a consequence, corners of algebraic Cuntz–Krieger algebras are algebraic Cuntz–Krieger algebras.

For a finite quiver Q without sources, we consider the corresponding radical square zero algebra A. We construct an explicit compact generator for the homotopy category of acyclic complexes of projective A-modules. We call such a generator the projective Leavitt complex of Q. This terminology is justified by the following result: the opposite differential graded endomorphism algebra of the projective Leavitt complex of Q is quasi-isomorphic to the Leavitt path algebra of Qop. Here, Qop is the opposite quiver of Q, and the Leavitt path algebra of Qop is naturally ${\open Z}$-graded and viewed as a differential graded algebra with trivial differential.

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 cancelative. We introduce the nonunital generalizations of graded analogs 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.

We show that every subset of vertices of a directed graph $E$ gives a Morita equivalence between a subalgebra and an ideal of the associated Leavitt path algebra. We use this observation to prove an algebraic version of a theorem of Crisp and Gow: certain subgraphs of $E$ can be contracted to a new graph $G$ such that the Leavitt path algebras of $E$ and $G$ are Morita equivalent. We provide examples to illustrate how desingularising a graph, and in- or out-delaying of a graph, all fit into this setting.

We continue to investigate branching systems of directed graphs and their connections with graph algebras. We give a sufficient condition under which the representation induced from a branching system of a directed graph is faithful and construct a large class of branching systems that satisfy this condition. We finish the paper by providing a proof of the converse of the Cuntz–Krieger uniqueness theorem for graph algebras by means of branching systems.

In Hazrat gave a K-theoretic invariant for Leavitt path algebras as graded algebras. Hazrat conjectured that this invariant classifies Leavitt path algebras up to graded isomorphism, and proved the conjecture in some cases. In this paper, we prove that a weak version of the conjecture holds for all finite essential graphs.

The stable rank of Leavitt path algebras of row-finite graphs was computed by Ara and Pardo. In this paper we extend this to an arbitrary directed graph. In part our computation proceeds as for the row-finite case, but we also use knowledge of the row-finite setting by applying the desingularising method due to Drinen and Tomforde. In particular, we characterise purely infinite simple quotients of a Leavitt path algebra.