Gene Abrams

Isomorphism and Morita equivalence of graph algebras

For any countable graph E, we investigate the relationship between the Leavitt path algebra L ℂ (E) and the graph C * -algebra C * (E). For graphs E and F, we examine ring homomorphisms, ring *-homomorphisms, algebra homomorphisms, and algebra *-homomorphisms between L ℂ (E) and L ℂ (F). We prove that in certain situations isomorphisms between L ℂ (E) and L ℂ (F) yield *-isomorphisms between the corresponding C * -algebras C * (E) and C * (F). Conversely, we show that *-isomorphisms between C * (E) and C * (F) produce isomorphisms between L ℂ (E) and L ℂ (F) in specific cases. The relationship between Leavitt path algebras and graph C * -algebras is also explored in the context of Morita equivalence.

Chain conditions for Leavitt path algebras

Abstract In this paper we give necessary and sufficient conditions on a row-finite graph E so that the corresponding (not necessarily unital) Leavitt path K-algebra LK (E) is either artinian or noetherian from both a local and a categorical perspective. These extend the known results in the unital case to a much wider context. Besides the graph theoretic conditions, we provide in both situations isomorphisms between these algebras and appropriate direct sums of matrix rings over K or K[x, x –1].

Generalizations, Applications, and Current Lines of Research

We conclude the book with various observations regarding three important aspects of Leavitt path algebras. First, we describe various generalizations of, and constructions related to, Leavitt path algebras. Next, we present some applications of Leavitt path algebras (specifically, we give some examples of results from outside the subject of Leavitt path algebras per se which have been established using the machinery developed for Leavitt path algebras). Finally, we consider some still-unresolved questions of interest.

General Ring-Theoretic Results

In this chapter we provide descriptions of Leavitt path algebras satisfying various well-studied ring-theoretic properties. These include: primeness and primitivity; chain conditions on one-sided ideals; self-injectivity; and the stable rank.

Two-Sided Ideals

In this chapter we investigate the ideal structure of Leavitt path algebras. We start by describing the natural (mathbb{Z})-grading on L K (E). We then present the Reduction Theorem; this result describes how elements of L K (E) may be transformed in some specified way to either a vertex or a cycle without exits. Numerous consequences are discussed, including the Uniqueness Theorems. We then establish in the Structure Theorem for Graded Ideals a precise relationship between graded ideals and explicit sets of idempotents (arising from hereditary and saturated subsets of vertices, together with breaking vertices). With this description of the graded ideals having been achieved, we focus in the remainder of the chapter on the structure of all ideals. We achieve in the Structure Theorem for Ideals an explicit description of the entire ideal structure of L K (E) (including both the graded and non-graded ideals) for an arbitrary graph E and field K. This result utilizes the Structure Theorem for Graded Ideals together with the analysis of the ideal generated by vertices which lie on cycles having no exits. A number of ring-theoretic results follow almost immediately from the Structure Theorem for Ideals, including the Simplicity Theorem. Along the way, we describe the socle of a Leavitt path algebra, and we achieve a description of the finite dimensional Leavitt path algebras.

Graph C ∗ -Algebras, and Their Relationship to Leavitt Path Algebras

In this chapter we investigate the connections between Leavitt path algebras (with coefficients in (mathbb{C})), and their analytic counterparts, the graph C ∗-algebras. We start by giving a brief overview of graph C ∗-algebras, and then show how the Leavitt path algebra (L_{mathbb{C}}(E)) naturally embeds as a dense ∗-subalgebra of the graph C ∗-algebra C ∗(E). We analyze the structure of the closed ideals in C ∗(E) for row-finite graphs, and compare this structure to the ideal structure of the corresponding Leavitt path algebra L K (E). We finish the chapter by considering numerous properties which are simultaneously shared by C ∗(E) and (L_{mathbb{C}}(E)).