Non-stable $K$-theory for Leavitt path algebras

Hay, Damon M.; Loving, Marissa; Montgomery, Martin; Ruiz, Efren; Todd, Katherine

doi:10.1216/rmj-2014-44-6-1817

Cited by 12 publications

(5 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The computation above then follows from the row-finite case and [13, Lemma 2.3]. Moreover, the claim about the positive cone of K alg 0 is established in [17,Theorem 4.9]. Also, since K is a field, we have (K alg 1 (K), +) ∼ = (K × , ·).…”

mentioning

confidence: 89%

Classification of unital simple Leavitt path algebras of infinite graphs

Ruiz

Tomforde

2013

Journal of Algebra

Self Cite

View full text Add to dashboard Cite

We prove that if E and F are graphs with a finite number of vertices and an infinite number of edges, if K is a field, and if LK (E) and LK (F ) are simple Leavitt path algebras, then LK (E) is Morita equivalent to LK (F ) if and only if K alg 0 (LK (E)) ∼ = K alg 0 (LK (F )) and the graphs E and F have the same number of singular vertices, and moreover, in this case one may transform the graph E into the graph F using basic moves that preserve the Morita equivalence class of the associated Leavitt path algebra. We also show that when K is a field with no free quotients, the condition that E and F have the same number of singular vertices may be replaced by K alg 1 (LK(E)) ∼ = K alg 1 (LK (F )), and we produce examples showing this cannot be done in general. We describe how we can combine our results with a classification result of Abrams, Louly, Pardo, and Smith to get a nearly complete classification of unital simple Leavitt path algebras -the only missing part is determining whether the "sign of the determinant condition" is necessary in the finite graph case. We also consider the Cuntz splice move on a graph and its effect on the associated Leavitt path algebra.

show abstract

mentioning

confidence: 89%

Classification of unital simple Leavitt path algebras of infinite graphs

Ruiz

Tomforde

2013

Journal of Algebra

Self Cite

View full text Add to dashboard Cite

show abstract

“…Hay et. al [23] proved the following interesting result. where V and W are finite subsets of E 0 , each v ∈ V is an infinite emitter, each T v is a nonempty finite subset of s −1 (v), and the numbers n v , n w are positive integers.…”

Section: Corners Of Graph C * -Algebrasmentioning

confidence: 90%

“…Then M E is defined to be the quotient monoid T / ∼ E ; we denote an element of M E by [x], where x ∈ T . The foundational result about Leavitt path algebras for our work is the following: 12,Theorem 4.3] and [23,Theorem 4.9]). Let E be an arbitrary graph and K any field.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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

Abrams

Ara

Molina

2017

Lecture Notes in Mathematics

View full text Add to dashboard Cite

show abstract

“…A quiver are quadruple of the sets which elements are called vertices, the set which elements are called arrows, and two maps r , which associate to each arrow , its source , and its range , respectively. Many studies have been carried out on Leavitt path algebra (Abrams et al, 2015;Abrams & Aranda Pino, 2005;Ara & Pardo, 2017;Ara & Rangaswamy, 2014;Arnone & Cortiñas, 2022;Clark et al, 2016;Hay et al, 2014;Rangaswamy, 2015). Abrams, et al in 2007 have studied the necessary and sufficient conditions for a quiver so that the Leavitt path algebra is a finitedimensional algebra.…”

Section: Introductionmentioning

confidence: 99%