Regularity Conditions for Arbitrary Leavitt Path Algebras

Abstract

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: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.

FP-injective semirings, semigroup rings and Leavitt path algebras

“…To reach a contradiction, we essentially follow the ideas in the proof of (2)=⇒ (3) in [4,Theorem 1], by expanding the above equation, considering it as a graded equation and comparing the degree of the components on both sides. Existence of terms of higher degrees on the left hand side with no terms of equal degree on the right hand side leads to a contradiction.…”

Weakly Regular and Self-Injective Leavitt Path Algebras Over Arbitrary Graphs

“…Hence L K (E) is von Neumann regular. By [7,Theorem 1] this is equivalent to the graph E being acyclic.…”

“…Although their history is very recent, a flurry of activity has followed since the beginning of the theory, in several different directions: characterization of algebraic properties of L K (E) in terms of graph properties of E (see for instance [3,4,6,18,23]); study of the modules over L K (E) in [14,21] among others; computation of various substructures such as the Jacobson radical, the center, the socle and the singular ideal in [5,19,21,42] respectively; investigation of the relationship and connections with C * (E) and general C*-algebras [9,17,20,24]; classification programs [1,2,10]; K-Theory [14,15,17]; and generalization to arbitrary graphs, which has been done in two stages, first for countable but not necessarily row-finite graphs in [5,22,43], and then for completely arbitrary graphs in [7,8,32].…”

“…We focus on the characterization of algebraic-theoretic properties of L K (E) in terms of graph-theoretic properties of E: the main objective is to keep on exploring regularity conditions for L K (E), thus extending the characterization of von Neumann regular Leavitt path algebras of [7]. Concretely, we completely determine the weakly regular and self-injective Leavitt path algebras in Theorem 3.15 and Theorem 4.7 respectively.…”

On Locally Unit Regularity Conditions for Arbitrary Leavitt Path Algebras

“…In [3], G. Abrams and K. M. Rangaswamy showed how de…nition of von Neumann regular ring (recall that a ring R is von Neumann regular if for every a 2 R there exists b 2 R such that a = aba) is extended to locally unit regular ring and in [3,Theorem 2] if is arbitrary graph, L K ( ) is locally unit regular if and only if is acyclic. This article is organized as follows.…”