Let T be a set of elements in a ring A. The set T is right permutable if for any a ∈ A and t ∈ T, there exist b ∈ A, u ∈ T such that au = tb. A multiplicative set in a ring A is any subset T of A such that 1 ∈ T,0 ∉ T and T is closed under multiplication. A completely prime ideal in a ring A is any proper ideal B such that A\B is a multiplicative set (i.e. A/Bis a domain). A minimal prime ideal (resp. minimal completely prime ideal) in a ring A is any prime (resp. completely prime) ideal P such that P contains no properly any other prime ideal (resp. completely prime ideal) of A. Let I be any proper ideal of a ring A. The set of all elements a ∈ A such that a + I is a regular element of A/I is denoted by c(I). In particular, c(0) is the set of all regular elements of A.

Rings of quotients

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.

/pdf/leavitt-path-algebras-584irhft0j.pdf

Leavitt Path Algebras

Uniqueness theorems and ideal structure for Leavitt path algebras

We extend the notion of the Leavitt path algebra of a graph to include all directed graphs. We show how various ring-theoretic properties of these more general structures relate to the corresponding properties of Leavitt path algebras of row-finite graphs. Specifically, we identify those graphs for which the corresponding Leavitt path algebra is simple; purely infinite simple; exchange; and semiprime. In our final result, we show that all Leavitt path algebras have zero Jacobson radical.

The Leavitt path algebras of arbitrary graphs

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 simplicity. We also use these results to give a proof of the fact that for any graph E the Leavitt path algebra $L_\mathbb{C}(E)$ embeds as a dense *-subalgebra of the graph C*-algebra C*(E). This embedding has consequences for graph C*-algebras, and we discuss how we obtain new information concerning the construction of C*(E).

Uniqueness Theorems and Ideal Structure for Leavitt Path Algebras

Exchange Leavitt path algebras and stable rank

Finite-dimensional Leavitt path algebras

A group-graded K-algebra A = ⊕
 g∈G
 A
 g
 is called locally finite in case each graded component A
 g
 is finite dimensional over K. We characterize the graphs E for which the Leavitt path algebra L
 K
 (E) is locally finite in the standard ℤ-grading. For a locally finite ℤ-graded algebra A we show that, if every nonzero graded ideal has finite codimension in A, then every nonzero ideal has finite codimension in A; that is, ℤ-graded just infinite implies just infinite. We use this result to characterize the finite graphs E for which the Leavitt path algebra L
 K
 (E) is locally finite just infinite. We then give an explicit description of the graphs and algebras which arise in this way. In particular, we show that the locally finite Leavitt path algebras are precisely the noetherian Leavitt path algebras.

Locally finite Leavitt path algebras

http://webpersonal.uma.es/~MSILESM/Pub_Papers_files/S29QPA.pdf

Algebras of quotients of path algebras

A notion of generalized inverse extending that of Moore—Penrose inverse for continuous linear operators between Hilbert spaces and that of group inverse for elements of an associative algebra is defined in any Jordan triple system (J, P). An element a∊J has a (unique) generalized inverse if and only if it is strongly regular, i.e., a∊P(a)2J. A Jordan triple system J is strongly regular if and only if it is von Neumann regular and has no nonzero nilpotent elements. Generalized inverses have properties similar to those of the invertible elements in unital Jordan algebras. With a suitable notion of strong associativity, for a strongly regular element a∊J with generalized inverse b the subtriple generated by {a, b} is strongly associative

http://atarazanas.sci.uma.es/docs/articulos/16737404.pdf

M. Siles Molina

Papers

Exchange Leavitt path algebras and stable rank

Finite-dimensional Leavitt path algebras

Locally finite Leavitt path algebras

Algebras of quotients of path algebras

Strong regularity and generalized inverses in jordan systems