Graph Algebras

Raeburn, Iain

doi:10.1090/cbms/103

Cited by 319 publications

(414 citation statements)

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“…We recall some terminology for graphs (see, for example, [12,2]; note that our edge direction convention agrees with that used in these papers and in [11], which is opposite to that used in [14]). …”

Section: Paths and Condition (K)mentioning

confidence: 99%

“…, n k } and l(α) = k. Then one can verify that β z (t α ) = z l(α) t α . It follows (see, for example, the argument of [14,Corollary 3.3]) that the fixed point algebra C * (E) β of the action β satisfies…”

Section: Gauge-invariant Idealsmentioning

confidence: 99%

“…This theory has been explored extensively in many papers (see [12,13,2,8] for seminal results, and [14] for a survey). Graph C * -algebras include many C * -algebras besides the Cuntz-Krieger algebras; in particular, graph C * -algebras include the Toeplitz algebra, continuous functions on the circle, all finite-dimensional C * -algebras, many AF-algebras, many purely infinite simple C * -algebras, and many Type I C * -algebras.…”

Section: Introductionmentioning

confidence: 99%

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Graph algebras, Exel-Laca algebras, and ultragraph algebras coincide up to Morita equivalence

Katsura¹,

Muhly²,

Sims³

et al. 2010

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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Abstract. We prove that the classes of graph algebras, Exel-Laca algebras, and ultragraph algebras coincide up to Morita equivalence. This result answers the long-standing open question of whether every Exel-Laca algebra is Morita equivalent to a graph algebra. Given an ultragraph G we construct a directed graph E such that C * (G) is isomorphic to a full corner of C * (E). As applications, we characterize real rank zero for ultragraph algebras and describe quotients of ultragraph algebras by gauge-invariant ideals.

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Section: Paths and Condition (K)mentioning

confidence: 99%

Section: Gauge-invariant Idealsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Graph algebras, Exel-Laca algebras, and ultragraph algebras coincide up to Morita equivalence

Katsura¹,

Muhly²,

Sims³

et al. 2010

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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“…For a recent survey of these algebras we point the reader to the notes [25]. Both O(E) and T(E) can be described either as universal objects or concretely.…”

Section: Or O(e)) and Its Toeplitz Extension T(e)mentioning

confidence: 99%

“…Our investigations draw on numerous aspects of contemporary operator algebra theory. We make use of fundamental results from the theory of graph C*-algebras [25] and Cuntz-Pimsner algebras [9,12,17,20,23]. The theory of C*-algebras associated with 'topological graphs', introduced by the second author and Muhly [19] and studied further in [11,12,13,20], plays a central role.…”

Section: Introductionmentioning

confidence: 99%

A Class of Limit Algebras Associated with Directed Graphs

Kribs¹,

Solel²

2007

J. Aust. Math. Soc.

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Every directed graph defines a Hilbert space and a family of weighted shifts that act on the space. We identify a natural notion of periodicity for such shifts and study their C -algebras. We prove the algebras generated by all shifts of a fixed period are of Cuntz-Krieger and Toeplitz-Cuntz-Krieger type. The limit C*-algebras determined by an increasing sequence of positive integers, each dividing the next, are proved to be isomorphic to Cuntz-Pimsner algebras and the linking maps are shown to arise as factor maps. We derive a characterization of simplicity and compute the tf-groups for these algebras. We prove a classification theorem for the class of algebras generated by simple loop graphs.2000 Mathematics subject classification: primary 46L05,47L40.

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On Semicircular Elements Induced by Connected Finite Graphs

Cho

Jørgensen

2023

Mathematics Online First Collections

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Graph Algebras

Cited by 319 publications

References 0 publications

Graph algebras, Exel-Laca algebras, and ultragraph algebras coincide up to Morita equivalence

Graph algebras, Exel-Laca algebras, and ultragraph algebras coincide up to Morita equivalence

A Class of Limit Algebras Associated with Directed Graphs

On Semicircular Elements Induced by Connected Finite Graphs

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