Traces on Semigroup Rings and Leavitt Path Algebras

2015

Algebr Represent Theor

Self Cite

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.

Section: Positive Faithful and Canonical Traces On Leavitt Path Algmentioning

confidence: 95%

“…To show the first sentence, consider [12,Proposition 19] proving that any map δ on G E ∪ {0} which preserves zero is central if and only if the following three conditions hold:…”

Section: Positive Faithful and Canonical Traces On Leavitt Path Algmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Canonical Traces and Directly Finite Leavitt Path Algebras

2015

Algebr Represent Theor

Self Cite

Baer and Baer *-ring characterizations of Leavitt path algebras

Hazrat

Journal of Pure and Applied Algebra

2018

Self Cite

Abstract. We characterize Leavitt path algebras which are Rickart, Baer, and Baer * -rings in terms of the properties of the underlying graph. In order to treat non-unital Leavitt path algebras as well, we generalize these annihilator-related properties to locally unital rings and provide a more general characterizations of Leavitt path algebras which are locally Rickart, locally Baer, and locally Baer * -rings. Leavitt path algebras are also graded rings and we formulate the graded versions of these annihilator-related properties and characterize Leavitt path algebras having those properties as well.Our characterizations provide a quick way to generate a wide variety of examples of rings. For example, creating a Baer and not a Baer * -ring, a Rickart * -ring which is not Baer, or a Baer and not a Rickart * -ring, is straightforward using the graph-theoretic properties from our results. In addition, our characterizations showcase more properties which distinguish behavior of Leavitt path algebras from their C * -algebra counterparts. For example, while a graph C * -algebra is Baer (and a Baer * -ring) if and only if the underlying graph is finite and acyclic, a Leavitt path algebra is Baer if and only if the graph is finite and no cycle has an exit, and it is a Baer * -ring if and only if the graph is a finite disjoint union of graphs which are finite and acyclic or loops.

Graded Chain Conditions and Leavitt Path Algebras of No-Exit Graphs

2017

J. Aust. Math. Soc.

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Abstract. We obtain a complete structural characterization of Cohn-Leavitt algebras over no-exit objects as graded involutive algebras. Corollaries of this result include graph-theoretic conditions characterizing when a Leavitt path algebra is a directed union of (graded) matricial algebras over the underlying field and over the algebra of Laurent polynomials and when the monoid of isomorphism classes of finitely generated projective modules is atomic and cancellative.We introduce the non-unital generalizations of graded analogues of noetherian and artinian rings, graded locally noetherian and graded locally artinian rings, and characterize graded locally noetherian and graded locally artinian Leavitt path algebras without any restriction on the cardinality of the graph. As a consequence, we relax the assumptions of the Abrams-Aranda-Perera-Siles characterization of locally noetherian and locally artinian Leavitt path algebras.