Representations of relative Cohn path algebras

Canto, Cristóbal Gil

doi:10.1016/j.jpaa.2020.106310

Cited by 10 publications

(12 citation statements)

References 28 publications

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“…Definition 2. 7. [22,Definition 3.2] Let Λ be a row-finite source-free k-graph and let (X, µ) be a measure space.…”

Section: The Radon-nikodym Derivative φmentioning

confidence: 99%

Irreducibility and monicity for representations of $k$-graph $C^*$-algebras

Farsi¹,

Gillaspy²

2021

Preprint

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The representations of a k-graph C * -algebra C * (Λ) which arise from Λ-semibranching function systems are closely linked to the dynamics of the k-graph Λ. In this paper, we undertake a systematic analysis of the question of irreducibility for these representations. We provide a variety of necessary and sufficient conditions for irreducibility, as well as a number of examples indicating the optimality of our results. We also explore the relationship between irreducible Λ-semibranching representations and purely atomic representations of C * (Λ). Throughout the paper, we work in the setting of rowfinite source-free k-graphs; this paper constitutes the first analysis of Λ-semibranching representations at this level of generality.

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“…Definition 2. 7. [22,Definition 3.2] Let Λ be a row-finite source-free k-graph and let (X, µ) be a measure space.…”

Section: The Radon-nikodym Derivative φmentioning

confidence: 99%

Irreducibility and monicity for representations of $k$-graph $C^*$-algebras

Farsi¹,

Gillaspy²

2021

Preprint

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“…In the previous sections, we studied in detail the Z-grading on L R (G). Another interesting grading on L R (G) is determined by the free group of the edges, F. This grading has been defined in [14], generalizing previous work on Leavitt path algebras of graphs [6,13,16], and it has been used to obtain interesting results on ultragraph (and graph) Leavitt path algebras, see [6,12,14] for example. All the results in this section are also new for Leavitt path algebras of graphs.…”

Section: Epsilon Strong Z-grading On L R (G)mentioning

confidence: 99%

“…In particular, the integer grading of a Leavitt path algebra has been studied in [20], where the unital Leavitt path algebras which are strongly graded are completely characterized, in [10,18], where strongly graded Leavitt path algebras are characterized in terms of Condition (Y), and in [23], where it is shown that the Leavitt path algebra associated to a finite graph is epsilon-strongly Z-graded. The grading over the free group of the edges has been introduced in [13] and has been used to give alternative proofs of interesting results, such as the Reduction Theorem and the simplicity criteria for Leavitt path algebras (see [13,16,12]) and is related to branching systems, see [6].…”

Section: Introductionmentioning

confidence: 99%

Properties of the gradings on ultragraph algebras via the underlying combinatorics

Royer¹

2021

Preprint

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There are two established gradings on Leavitt path algebras associated with ultragraphs, namely the grading by the integers group and the grading by the free group on the edges. In this paper, we characterize properties of these gradings in terms of the underlying combinatorial properties of the ultragraphs. More precisely, we characterize when the gradings are strong or epsilon-strong. The results regarding the free group on the edges are new also in the context of Leavitt path algebras of graphs. Finally, we also describe the relation between the strongness of the integer grading on an ultragraph Leavitt path algebra and the saturation of the gauge action associated with the corresponding ultragraph C*-algebra.

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“…Iterated function systems and branching systems are widely used in the study of representations of algebras associated to combinatorial objects, see for example [8,11,12,13,14,15,16,17,18,19,20,21,23,25,31]. Hence it is interesting to note that the representation π of Theorem 3.7 can be constructed via branching systems.…”

Section: Branching Systemsmentioning

confidence: 99%

“…Remark 4.3 The above branching system can also be seen as a partial action of the free group on the edges of the ultragraph and be used to realized L R (G) as a partial skew group ring, see [8,22,24].…”

Section: Branching Systemsmentioning

confidence: 99%

Irreducible and permutative representations of ultragraph Leavitt path algebras

Royer

2019

Forum Mathematicum

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We completely characterize perfect, permutative, irreducible representations of an ultragraph Leavitt path algebra. For this we extend to ultragraph Leavitt path algebras Chen's construction of irreducible representations of Leavitt path algebras. We show that these representations can be built from branching system and characterize irreducible representations associated to perfect branching systems. Along the way we improve the characterization of faithfulness of Chen's irreducible representations.

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Representations of relative Cohn path algebras

Cited by 10 publications

References 28 publications

Irreducibility and monicity for representations of $k$-graph $C^*$-algebras

Irreducibility and monicity for representations of $k$-graph $C^*$-algebras

Properties of the gradings on ultragraph algebras via the underlying combinatorics

Irreducible and permutative representations of ultragraph Leavitt path algebras

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