Uniqueness theorems and ideal structure for Leavitt path algebras

Tomforde, Mark

doi:10.1016/j.jalgebra.2007.01.031

Cited by 163 publications

(316 citation statements)

References 24 publications

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“…The following theorem follows from results by K. R. Goodearl [32] and M. Tomforde [43] and will be of importance to characterize right weakly regular Leavitt path algebras. Theorem 3.4.…”

Section: Weakly Regular Leavitt Path Algebrasmentioning

confidence: 86%

“…According to [43,Theorem 5.7], all the graded ideals of a Leavitt path algebra L K (E) are of the form I (H,S) for a suitable pair (H, S), being H a hereditary and saturated subset of vertices of E 0 .…”

Section: Proof (Ii) ⇔ (Iii) Is [32 Theorem 42] While (I) ⇔ (Iii) mentioning

confidence: 99%

“…Even though a great amount of algebraic properties of L K (E) have been characterized in terms of properties of the graph (and, depending on the type of graphs considered, an overview of these works are: [3, 4, 6, 17, 18, ?, 23] for the row-finite case, [5,22,42] for the countable but not necessarily row-finite setting, and [8,43,32] for arbitrary graphs), this is the first instance where, starting from an arbitrary graph E, an algebraic property of L K (E) automatically implies row-finiteness of the graph E.…”

Section: Self-injective Leavitt Path Algebrasmentioning

confidence: 99%

“…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].…”

Section: Introductionmentioning

confidence: 99%

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

Pino¹,

Rangaswamy

Molina³

2010

Algebr Represent Theor

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“…The following theorem follows from results by K. R. Goodearl [32] and M. Tomforde [43] and will be of importance to characterize right weakly regular Leavitt path algebras. Theorem 3.4.…”

Section: Weakly Regular Leavitt Path Algebrasmentioning

confidence: 86%

Section: Proof (Ii) ⇔ (Iii) Is [32 Theorem 42] While (I) ⇔ (Iii) mentioning

confidence: 99%

Section: Self-injective Leavitt Path Algebrasmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

Pino¹,

Rangaswamy

Molina³

2010

Algebr Represent Theor

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“…Thus φ is a monomorphism by the Graded Uniqueness Theorem ( [15,Theorem 4.8]). Note that E is assumed to be countable in [15] (1) The involution on K is positive definite.…”

Section: Cohn-leavitt Algebras and Directly Finite Leavitt Path Algebrasmentioning

confidence: 99%

Canonical Traces and Directly Finite Leavitt Path Algebras

Vaš

2015

Algebr Represent Theor

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Abstract. Motivated by the study of traces on graph C * -algebras, we consider traces (additive, central maps) on Leavitt path algebras, the algebraic counterparts of graph C * -algebras. In particular, we consider traces which vanish on nonzero graded components of a Leavitt path algebra and refer to them as canonical since they are uniquely determined by their values on the vertices.A desirable property of a C-valued trace on a C * -algebra is that the trace of an element of the positive cone is nonnegative. We adapt this property to traces on a Leavitt path algebra L K (E) with values in any involutive ring. We refer to traces with this property as positive. If a positive trace is injective on positive elements, we say that it is faithful. We characterize when a canonical, K-linear trace is positive and when it is faithful in terms of its values on the vertices. As a consequence, we obtain a bijective correspondence between the set of faithful, gauge invariant, C-valued (algebra) traces on L C (E) of a countable graph E and the set of faithful, semifinite, lower semicontinuous, gauge invariant (operator theory) traces on the corresponding graph C * -algebra C * (E).With the direct finite condition (i.e xy = 1 implies yx = 1) for unital rings adapted to rings with local units, we characterize directly finite Leavitt path algebras as exactly those having the underlying graphs in which no cycle has an exit. Our proof involves consideration of "local" Cohn-Leavitt subalgebras of finite subgraphs. Lastly, we show that, while related, the class of locally noetherian, the class of directly finite, and the class of Leavitt path algebras which admit a faithful trace are different in general.

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