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Johansen, Rune; Sørensen, Adam P. W.

doi:10.1016/j.jpaa.2016.05.023

Cited by 17 publications

(17 citation statements)

References 36 publications

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“…Hence, Theorem 1.4, in particular, algebraically differentiate Leavitt path algebras with coefficients in C respectively Z. The author feels that this differentiaton is especially interesting considering the strange behaviour of Leavitt path algebras with coefficients in Z observed by Johansen and Sørensen[18].…”

mentioning

confidence: 79%

“…Tomforde [26] introduced Leavitt path algebras over commutative unital rings and proved that many results carry over to his generalized setting. Leavitt path algebras over Z were considered by Johansen and Sørensen [18] in connection to the classification program of Leavitt path algebras. In this article, we follow Hazrat [17] and consider Leavitt path algebras L R (E) where R is a general, possibly non-commutative associative ring.…”

Section: Introductionmentioning

confidence: 99%

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Chain Conditions for Epsilon-Strongly Graded Rings with Applications to Leavitt Path Algebras

Lännström

2019

Algebr Represent Theor

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We provide a characterization of graded von Neumann regular rings involving the recently introduced class of nearly epsilon-strongly graded rings. As our main application, we generalize Hazrat's result that Leavitt path algebras over fields are graded von Neumann regular. More precisely, we show that a Leavitt path algebra LR(E) with coefficients in a unital ring R is graded von Neumann regular if and only if R is von Neumann regular. We also prove that both Leavitt path algebras and corner skew Laurent polynomial rings over von Neumann regular rings are semiprimitive and semiprime. Thereby, we generalize a result by Abrams and Aranda Pino on the semiprimitivity of Leavitt path algebras over fields.

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mentioning

confidence: 79%

Section: Introductionmentioning

confidence: 99%

Chain Conditions for Epsilon-Strongly Graded Rings with Applications to Leavitt Path Algebras

Lännström

2019

Algebr Represent Theor

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“…Since every unit in O n1 × · · · × O n k has isotropy group Z k and every unit in O m1 × · · · × O m l has isotropy group Z l , deduce that k = l. The adjacency matrix of the directed graph with one vertex and n edges is the 1 × 1 matrix [n]. The Bowen-Franks group associated to [n] is defined (see [25,28]) as BF([n]) = Z/(n − 1)Z. Therefore, BF([n k ]) ∼ = BF([m k ]) if and only if n k = m k .…”

Section: Tensor Products Of Leavitt Algebrasmentioning

confidence: 99%

Tensor Products of Steinberg Algebras

Rigby¹

2019

J. Aust. Math. Soc.

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We prove that A R (G) ⊗ R A R (H) ∼ = A R (G × H), if G and H are Hausdorff ample groupoids. As part of the proof, we give a new universal property of Steinberg algebras. We then consider the isomorphism problem for tensor products of Leavitt algebras, and show that no diagonal-preserving isomorphism exists between L 2,R ⊗ L 3,R and L 2,R ⊗ L 2,R . Indeed, there are no unexpected diagonalpreserving isomorphisms between tensor products of finitely many Leavitt algebras. We give an easy proof that every * -isomorphism of Steinberg algebras over the integers preserves the diagonal, and it follows that L 2,Z ⊗ L 3,Z ∼ = L 2,Z ⊗ L 2,Z (as * -rings).

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“…Even combinations of these restrictions (i.e., restricting both the graph and the field) yield open problems -no one knows, for example, whether L C (E 2 ) and L C (E − 2 ) are Morita equivalent, or whether L Z 2 (E 2 ) and L Z 2 (E − 2 ) are Morita equivalent. In July 2015, Johansen and Sørensen announced the preprint [11], which to the author's knowledge contain some of the first concrete results concerning the sign of the determinant condition. Although Leavitt path algebras are defined over fields, as noted by the author in [22], for any graph E and any commutative ring R it is possible to construct a Leavitt path algebra L R (E) with coefficients in R. Johansen and Sørensen proved that if we choose the coefficients to be the ring Z, then…”

Section: Classification Of Leavitt Path Algebras Of Finite Graphsmentioning

confidence: 99%

Classification of Graph Algebras: A Selective Survey

Tomforde

2016

Operator Algebras and Applications

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This survey reports on current progress of programs to classify graph C * -algebras and Leavitt path algebras up to Morita equivalence using K-theory. Beginning with an overview and some history, we trace the development of the classification of simple and nonsimple graph C * -algebras and state theorems summarizing the current status of these efforts. We then discuss the much more nascent efforts to classify Leavitt path algebras, and we describe the current status of these efforts as well as outline current impediments that must be solved for this classification program to progress. In particular, we give two specific open problems that must be addressed in order to identify the correct K-theoretic invariant for classification of simple Leavitt path algebras, and we discuss the significance of various possible outcomes to these open problems.

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The Cuntz splice does not preserve ⁎-isomorphism of Leavitt path algebras overZ

Cited by 17 publications

References 36 publications

Chain Conditions for Epsilon-Strongly Graded Rings with Applications to Leavitt Path Algebras

Chain Conditions for Epsilon-Strongly Graded Rings with Applications to Leavitt Path Algebras

Tensor Products of Steinberg Algebras

Classification of Graph Algebras: A Selective Survey

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