The central concept of this thesis is that of Leavitt path algebras, a notion introduced by both Abrams and Aranda Pino in [AA1] and Ara, Moreno and Pardo in [AMP] in 2004. The idea of using a field K and row-finite graph E to generate an algebra LK(E) provides an algebraic analogue to Cuntz and Krieger’s work with C∗-algebras of the form C∗(E) (which, despite the name, are analytic concepts). At the same time, Leavitt path algebras also generalise the algebras constructed by W. G. Leavitt in [Le1] and [Le2], and it is from this connection that the Leavitt path algebras get their name. Although the concept of a Leavitt path algebra is relatively new, in the years since the publication of [AA1] there has been a flurry of activity on the subject. Many results were initially shown for row-finite graphs, then extended to countable (but not necessarily row-finite) graphs (as in [AA3]) and then finally shown for completely arbitrary graphs (see, for example, [AR]). Most of the research has focused on the connections between ring-theoretic properties of LK(E) and graphtheoretic properties of E (for example [AA2], [AR] and [ARM2]), the socle and socle series of a Leavitt path algebra ([AMMS1], [AMMS2] and [ARM1]) and analogues between LK(E) and their C ∗-algebraic equivalents C∗(E) (for example [To]). Some papers have classified certain sets of Leavitt path algebras, such as [AAMMS], which classifies the Leavitt path algebras of graphs with up to three vertices (and without parallel edges). In Chapter 1 we will cover the ring-, moduleand graph-theoretic background necessary to examine these algebras in depth, as well as taking a brief look at Morita equivalence, a concept that will prove useful at various points in this thesis.

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Leavitt Path Algebras

The construction of the Leavitt path algebra associated to a directed graph $E$ is extended to incorporate a family $C$ consisting of partitions of the sets of edges emanating from the vertices of $E$. The new algebras, $L_K(E,C)$, are analyzed in terms of their homology, ideal theory, and K-theory. These algebras are proved to be hereditary, and it is shown that any conical abelian monoid occurs as the monoid $\mon{L_K(E,C)}$ of isomorphism classes of finitely generated projective modules over one of these algebras. The lattice of trace ideals of $L_K(E,C)$ is determined by graph-theoretic data, namely as a lattice of certain pairs consisting of a subset of $E^0$ and a subset of $C$. Necessary conditions for $\mon{L_K(E,C)}$ to be a refinement monoid are developed, together with a construction that embeds $(E,C)$ in a separated graph $(E_+,C^+)$ such that $\mon{L_K(E_+,C^+)}$ has refinement.

Leavitt path algebras of separated graphs

The algebraic structures known as Leavitt path algebras were initially developed in 2004 by Ara, Moreno and Pardo, and almost simultaneously (using a different approach) by the author and Aranda Pino. During the intervening decade, these algebras have attracted significant interest and attention, not only from ring theorists, but from analysts working in C ∗ -algebras, group theorists, and symbolic dynamicists as well. The goal of this article is threefold: to introduce the notion of Leavitt path algebras to the general mathematical community; to present some of the important results in the subject; and to describe some of the field’s currently unresolved questions.

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Leavitt path algebras: the first decade

A Leavitt path algebra associates to a directed graph a ℤ-graded algebra and in its simplest form it recovers the Leavitt algebra L(1, k). In this note, we first study this ℤ-grading and characterize the (ℤ-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 ⊕ℤ-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.

The graded structure of Leavitt path algebras

We show that if E is an arbitrary acyclic graph then the Leavitt path algebra LK(E) is locally K-matricial; that is, LK(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) LK(E) is von Neumann regular. (2) LK(E) is π-regular. (3) E is acyclic. (4) LK(E) is locally K-matricial. (5) LK(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.

Regularity Conditions for Arbitrary Leavitt Path Algebras

An introduction to Leavitt path algebras of arbitrary directed graphs is presented, and direct limit techniques are developed, with which many results that had previously been proved for countable graphs can be extended to uncountable ones. Such results include characterizations of simplicity, characterizations of the exchange property, and cancellation conditions for the K-theoretic monoid of equivalence classes of idempotent matrices.

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K. R. Goodearl

Papers

Leavitt path algebras and direct limits