*-Regular Leavitt path algebras of arbitrary graphs

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 positive-definite, even in the case when the graph E is not necessarily finite or row-finite. It has been shown that the Leavitt path algebra L _K (E) is regular if and only if E is acyclic. We give necessary and sufficient conditions for L _K (E) to be *-regular (i.e., regular with proper involution). This characterization of *-regularity of a Leavitt path algebra is given in terms of an algebraic property of K, not just a graph-theoretic property of E. This differs from the known characterizations of various other algebraic properties of a Leavitt path algebra in terms of graphtheoretic properties of E alone. As a corollary, we show that Handelman’s conjecture (stating that every *-regular ring is unit-regular) holds for Leavitt path algebras. Moreover, its generalized version for rings with local units also continues to hold for Leavitt path algebras over arbitrary graphs.     

The graded structure of Leavitt path algebras

A Leavitt path algebra associates to a directed graph a ?-graded algebra and in its simplest form it recovers the Leavitt algebra L(1, k). In this note, we first study this ?-grading and characterize the (?-graded) structure of Leavitt path algebras, associated to finite acyclic graphs, C _n -comet, multi-headed graphs and a mixture of these graphs (i.e., polycephaly graphs). The last two types are examples of graphs whose Leavitt path algebras are strongly graded. We give a criterion when a Leavitt path algebra is strongly graded and in particular characterize unital Leavitt path algebras which are strongly graded completely, along the way obtaining classes of algebras which are group rings or crossed-products. In an attempt to generalize the grading, we introduce weighted Leavitt path algebras associated to directed weighted graphs which have natural ⊕?-grading and in their simplest form recover the Leavitt algebras L(n, k). We then show that the basic properties of Leavitt path algebras can be naturally carried over to weighted Leavitt path algebras.     

Applications of Normal Forms for Weighted Leavitt Path Algebras: Simple Rings and Domains

Weighted Leavitt path algebras (wLpas) are a generalisation of Leavitt path algebras (with graphs of weight 1) and cover the algebras L _K (n, n + k) constructed by Leavitt. Using Bergman’s diamond lemma, we give normal forms for elements of a weighted Leavitt path algebra. This allows us to produce a basis for a wLpa. Using the normal form we classify the wLpas which are domains, simple and graded simple rings. For a large class of weighted Leavitt path algebras we establish a local valuation and as a consequence we prove that these algebras are prime, semiprimitive and nonsingular but contrary to Leavitt path algebras, they are not graded von Neumann regular.     

Finite-dimensional Leavitt path algebras

We classify the directed graphs E for which the Leavitt path algebra L(E) is finite dimensional. In our main results we provide two distinct classes of connected graphs from which, modulo the one-dimensional ideals, all finite-dimensional Leavitt path algebras arise.     

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.     

The structure of Leavitt path algebras and the Invariant Basis Number property

In this paper, we provide the structure of the Leavitt path algebra of a finite graph via some step-by-step process of source eliminations, and restate Kanuni and Özaydin's nice criterion for Leavitt path algebras of finite graphs having Invariant Basis Number via matrix-theoretic language. Consequently, we give a matrix-theoretic criterion for the Leavitt path algebra of a finite graph having Invariant Basis Number in terms of a sequence of source eliminations. Using these results, we show certain classes of finite graphs for which Leavitt path algebras have Invariant Basis Number, as well as investigate the Invariant Basis Number property of Leavitt path algebras of certain Cayley graphs of finite groups.     

Leavitt path algebras with coefficients in a commutative ring

Given a directed graph E we describe a method for constructing a Leavitt path algebra L_R(E) whose coefficients are in a commutative unital ring R. We prove versions of the Graded Uniqueness Theorem and Cuntz-Krieger Uniqueness Theorem for these Leavitt path algebras, giving proofs that both generalize and simplify the classical results for Leavitt path algebras over fields. We also analyze the ideal structure of L_R(E), and we prove that if K is a field, then L_K(E)≅K⊗_ZL_Z(E).     

The socle series of a Leavitt path algebra

We investigate the ascending Loewy socle series of Leavitt path algebras L _K (E) for an arbitrary graph E and field K. We classify those graphs E for which L _K (E) = S _λ for some element S _λ of the Loewy socle series. We then show that for any ordinal λ there exists a graph E so that the Loewy length of L _K (E) is λ. Moreover, λ ≤ ω ₁ (the first uncountable ordinal) if E is a row-finite graph.     

Gelfand–Kirillov Dimension of Algebras with a Locally Finite and Multiparameter Indexed Filtration

Let A have a locally finite and multiparameter indexed filtration ?, and let B be a homomorphic image of A. Thus B has the locally finite and multiparameter indexed filtration induced from ?. Here we study a relation between the associated graded algebra of A and that of B and use this result to calculate the Gelfand–Kirillov dimension of several algebras related to quantized algebras and Poisson enveloping algebras.     

Commutator Leavitt Path Algebras

For any field K and directed graph E, we completely describe the elements of the Leavitt path algebra L _K (E) which lie in the commutator subspace [L _K (E), L _K (E)]. We then use this result to classify all Leavitt path algebras L _K (E) that satisfy L _K (E)?=?[L _K (E),L _K (E)]. We also show that these Leavitt path algebras have the additional (unusual) property that all their Lie ideals are (ring-theoretic) ideals, and construct examples of such rings with various ideal structures.     

Piecewise-Koszul algebras

It is a small step toward the Koszul-type algebras.The piecewise-Koszul algebras are, in general,a new class of quadratic algebras but not the classical Koszul ones,simultaneously they agree with both the classical Koszul and higher Koszul algebras in special cases.We give a criteria theorem for a graded algebra A to be piecewise-Koszul in terms of its Yoneda-Ext algebra E(A),and show an A_∞-structure on E(A).Relations between Koszul algebras and piecewise-Koszul algebras are discussed.In particular,our results are related to the third question of Green-Marcos.     

Regularity Conditions for Arbitrary Leavitt Path Algebras

We show that if E is an arbitrary acyclic graph then the Leavitt path algebra L _K (E) is locally K-matricial; that is, L _K (E) is the direct union of subalgebras, each isomorphic to a finite direct sum of finite matrix rings over the field K. (Here an arbitrary graph means that neither cardinality conditions nor graph-theoretic conditions (e.g. row-finiteness) are imposed on E. These unrestrictive conditions are in contrast to the hypotheses used in much of the literature on this subject.) As a consequence we get our main result, in which we show that the following conditions are equivalent for an arbitrary graph E: (1) L _K (E) is von Neumann regular. (2) L _K (E) is π-regular. (3) E is acyclic. (4) L _K (E) is locally K-matricial. (5) L _K (E) is strongly π-regular. We conclude by showing how additional regularity conditions (unit regularity, strongly clean) can be appended to this list of equivalent conditions.     

The Injective Leavitt Complex

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

Bar-invariant bases of the quantum cluster algebra of type <Emphasis Type="Italic">A</Emphasis><Stack><Subscript>2</Subscript><Superscript>(2)</Superscript></Stack>

We construct bar-invariant ℤ[q ^±1/2]-bases of the quantum cluster algebra of the valued quiver A ₂⁽²⁾, one of which coincides with the quantum analogue of the basis of the corresponding cluster algebra discussed in P. Sherman, A. Zelevinsky: Positivity and canonical bases in rank 2 cluster algebras of finite and affine types, Moscow Math. J., 4, 2004, 947–974.     

The Dixmier-Moeglin Equivalence for Leavitt Path Algebras

Let K be a field, let E be a finite directed graph, and let L _K (E) be the Leavitt path algebra of E over K. We show that for a prime ideal P in L _K (E), the following are equivalent:

Graded central polynomials for the algebra <Emphasis Type="Italic">M</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>(<Emphasis Type="Italic">K</Emphasis>)

Let M _n (K) be the algebra of all n × n matrices over an infinite field K. This algebra has a natural ℤ _n -grading and a natural ℤ-grading. Finite bases for its ℤ _n -graded identities and for its ℤ-graded identities are known. In this paper we describe finite generating sets for the ℤ _n -graded and for the ℤ-graded central polynomials for M _n (K) Partially supported by CNPq 620025/2006-9     

The graded Grothendieck group and the classification of Leavitt path algebras

This paper is an attempt to show that, parallel to Elliott’s classification of AF C*-algebras by means of K-theory, the graded K ₀-group classifies Leavitt path algebras completely. In this direction, we prove this claim at two extremes, namely, for the class of acyclic graphs (graphs with no cycles) and multi-headed comets or rose graphs (graphs in which each head is connected to a cycle or to a collection of loops), or a mixture of these graphs (i.e., polycephaly graphs).     

Isometries between function algebras with finite codimensional range

Let A be a function algebra on a compact space X. A linear isometry T of A into A is said to be codimension n or finite codimensional if the range of T has codimension n in A. In this paper we prove that such isometries can be represented as weighted composition mappings on a cofinite subset, (∂A)₀, of the Shilov boundary for A, ∂A. We focus on those finite codimensional isometries for which (∂A)₀=∂A. All the above results, applied to the particular case of codimension 1 linear isometries on C(X), are used to improve the classification provided by Gutek et al. in J. Funct. Anal. 101, 97–119 (1991). Received: 3 June 1998 / Revised version: 22 March 1999