Equivalence and stable isomorphism of groupoids, and diagonal-preserving stable isomorphisms of graph $C^*$-algebras and Leavitt path algebras

<p>We investigate strongly graded C*-algebras. We focus on graph C*-algebras and explore the connection between graph C*-algebras and Leavitt path algebras, both of which are $\Z$-graded. It is known that a graphical condition called \emph{Condition (Y)} is necessary and sufficient for Leavitt path algebras to be strongly graded. In this thesis we prove this can be translated to the graph C*-algebra and prove that a graph C*-algebra associated to a row-finite graph is strongly graded if and only if Condition (Y) holds.</p>

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Realizing corners of Leavitt path algebras as Steinberg algebras, with corresponding connections to graph C⁎-algebras

Journal of Algebra ◽

10.1016/j.jalgebra.2021.11.004 ◽

2021 ◽

Author(s):

G. Abrams ◽

M. Dokuchaev ◽

T.G. Nam

Keyword(s):

C Algebras ◽

Path Algebras ◽

Leavitt Path Algebras

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A class of corners of a Leavitt path algebra

Science and Technology Development Journal - Natural Sciences ◽

10.32508/stdjns.v2i4.813 ◽

2019 ◽

Vol 2 (4) ◽

pp. 75-81

Author(s):

Deo Thanh Trinh

Keyword(s):

Directed Graph ◽

Finite Subset ◽

Path Algebra ◽

Graph Algebras ◽

Leavitt Path Algebra ◽

C Algebras ◽

Path Algebras ◽

Leavitt Path Algebras

Let E be a directed graph, K a field and LK(E) the Leavitt path algebra of E over K. The goal of this paper is to describe the structure of a class of corners of Leavitt path algebras LK(E). The motivation of this work comes from the paper “Corners of Graph Algebras” of Tyrone Crisp in which such corners of graph C*-algebras were investigated completely. Using the same ideas of Tyrone Crisp, we will show that for any finite subset X of vertices in a directed graph E such that the hereditary subset HE(X) generated by X is finite, the corner ( ) ( )( )     K v X v X v L E v is isomorphic to the Leavitt path algebra LK(EX) of some graph EX. We also provide a way how to construct this graph EX.

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Strongly Graded C*-algebras

10.26686/wgtn.17147585.v1 ◽

2021 ◽

Author(s):

◽

Ellis Dawson

Keyword(s):

Finite Graph ◽

C Algebra ◽

C Algebras ◽

Path Algebras ◽

Leavitt Path Algebras ◽

Necessary And Sufficient

<p>We investigate strongly graded C*-algebras. We focus on graph C*-algebras and explore the connection between graph C*-algebras and Leavitt path algebras, both of which are $\Z$-graded. It is known that a graphical condition called \emph{Condition (Y)} is necessary and sufficient for Leavitt path algebras to be strongly graded. In this thesis we prove this can be translated to the graph C*-algebra and prove that a graph C*-algebra associated to a row-finite graph is strongly graded if and only if Condition (Y) holds.</p>

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A note on the regular ideals of Leavitt path algebras

Journal of Algebra and Its Applications ◽

10.1142/s0219498822502255 ◽

2021 ◽

pp. 2250225

Author(s):

Daniel Gonçalves ◽

Danilo Royer

Keyword(s):

Finite Graph ◽

Path Algebra ◽

Leavitt Path Algebra ◽

Arbitrary Graph ◽

C Algebras ◽

Path Algebras ◽

Regular Ideal ◽

Vertex Set ◽

Leavitt Path Algebras

We show that, for an arbitrary graph, a regular ideal of the associated Leavitt path algebra is also graded. As a consequence, for a row-finite graph, we obtain that the quotient of the associated Leavitt path by a regular ideal is again a Leavitt path algebra and that Condition (L) is preserved by quotients by regular ideals. Furthermore, we describe the vertex set of a regular ideal and make a comparison between the theory of regular ideals in Leavitt path algebras and in graph C*-algebras.

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