The path space of a higher-rank graph

Webster, Samuel B. G.

doi:10.4064/sm204-2-4

Cited by 13 publications

(35 citation statements)

References 10 publications

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“…Since is finitely align, Theorem 4.1 of Webster [8] implies that is also finitely align. By the hypothesis, has no cycles, hence Proposition 4.4 of Rosjanuardi [2] implies that ) is finite dimensional.…”

Section: Propositionmentioning

confidence: 99%

“…By the hypothesis, has no cycles, hence Proposition 4.4 of Rosjanuardi [2] implies that ) is finite dimensional. From Proposition 6.6 of Webster [8] we get that is a subalgebra of ), and hence is finite dimensional.…”

Section: Propositionmentioning

confidence: 99%

“…There are several methods which put the graph in a bigger one so that the bigger one has no sources, for example is Farthing's desingularisation method [7], Webster's removing sources method [8], and desourcification method given by Clark, Flynn and an Huef [3]. These methods enable us to handle -graph with sources just like handling one without sources.…”

Section: Desourcificationmentioning

confidence: 99%

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Kumjian-Pask algebras of desourcification

Rosjanuardi¹,

Yusnitha²

2016

AIP Conference Proceedings

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Abstract. Kumjian-Pask algebra which was introduced by Pino, Clark, an Huef and Raeburn [1] in 2013, gives a purely algebraic version of a k-graph algebra. Rosjanuardi [2] gave necessary and sufficient condition of finitely dimensional complex Kumjian-Pask algebra of row-finite k-graph without sources. We will improve the previous results which allows us to deal with sources. We will consider Kumjian-Pask algebra for locally convex row-finite k-graph which was introduced by Clark, Flynn and an Huef [3], and use the desourcification of the graph to get conditions which characterise when the complex Kumjian-Pask algebra of locally convex row-finite k-graph is finite dimensional.

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Section: Propositionmentioning

confidence: 99%

Section: Propositionmentioning

confidence: 99%

Section: Desourcificationmentioning

confidence: 99%

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Kumjian-Pask algebras of desourcification

Rosjanuardi¹,

Yusnitha²

2016

AIP Conference Proceedings

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“…So, we may characterize the structure of primitive ideal space of C * (Λ) using that of C * (Λ). Note that the Farthing's desourcification was modified by Webster in [18,Section 4]. However, [18,Proposition 4.12] shows that the desourcifications constructed in [4,Section 3] and [18,Section 4] are isomorphic when Λ is row-finite.…”

Section: Introductionsmentioning

confidence: 99%

Primitive ideal space of higher-rank graph C⁎-algebras and decomposability

Larki

2019

Journal of Mathematical Analysis and Applications

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In this paper, we describe primitive ideal space of the C *algebra C * (Λ) associated to any locally convex row-finite k-graph Λ. To do this, we will apply the Farthing's desourcifying method on a recent result of Carlsen, Kang, Shotwell, and Sims. We also characterize certain maximal ideals of C * (Λ).Furthermore, we study the decomposability of C * (Λ). We apply the description of primitive ideals to show that if I is a direct summand of C * (Λ), then it is gauge-invariant and isomorphic to a certain k-graph C * -algebra. So, we may characterize decomposable higher-rank C *algebras by giving necessary and sufficient conditions for the underlying k-graphs. Moreover, we determine all such C * -algebras which can be decomposed into a direct sum of finitely many indecomposable C *algebras.

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“…In this article, we focus on row-finite k-graphs. For further discussion about row-finite k-graphs and their generalisations, see [24,30,31,32,40].…”

Section: Preliminariesmentioning

confidence: 99%

Cohn Path Algebras of Higher-Rank Graphs

Clark

Pangalela

2016

Algebr Represent Theor

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Abstract. In this article, we introduce Cohn path algebras of higher-rank graphs. We prove that for a higher-rank graph Λ, there exists a higher-rank graph T Λ such that the Cohn path algebra of Λ is isomorphic to the Kumjian-Pask algebra of T Λ. We then use this isomorphism and properties of Kumjian-Pask algebras to study Cohn path algebras. This includes proving a uniqueness theorem for Cohn path algebras. IntroductionLeavitt path algebras were introduced and studied in [3] and [9] as a generalisation the class of algebras studied by Leavitt in [25]. Leavitt path algebras are also the natural algebraic analogues of graph C * -algebras studied in [32]; a number of interesting result have been proven by translating between graph C * -algebras and the ring theoretic Leavitt path algebras.Cohn path algebras were introduced in [6,7] and generalise the algebras U 1,n studied by Cohn in [16]. The idea is to build an algebra out of the path space of a graph; addition and scalar multiplication are defined formally and multiplication of two paths is only nonzero when one can concatenate the paths. Cohn path algebras can also be obtained from Leavitt path algebras by omitting one of the Cuntz-Krieger relations. Hence every Leavitt path algebra can be viewed as a quotient of a Cohn path algebra. On the other hand, Abrams, Ara and Siles Molina show in [2] that for every graph E, there exists a graph T E (denoted E (X) in [2]) such that the Cohn path algebra of E is isomorphic to the Leavitt path algebra of T E.Although it has received less attention, the Cohn path algebra of a directed graph is the algebraic analogue of the C * -algebraic Toeplitz algebra of E as defined in [19]. In the C * -algebraic setting, Muhly and Tomforde in [27] and also Sims in [38] each show that for a graph E, the Toeplitz algebra of E is isomorphic to the graph C * -algebra of T E. Now we move into the setting of higher-rank graph algebras: In [24], Kumjian and Pask introduced a combinatorial model, called a higher-rank graph, in order to capture the essential features of the C * -algebras studied by Robertson and Steger in [35]. A higherrank graph, also called a k-graph, is a generalisation of the path category of a directed graph where the length of a 'path' λ in a k-graph is an element of N k . Kumjian and Pask studied the C * -algebras associated to row-finite higher-rank graphs with no sources. Raeburn, Sims and Yeend generalised Kumjian and Pask's construction by describing the class of C * -algebras associated to more general higher-rank graphs in [30,31].1991 Mathematics Subject Classification. 16S99 (Primary); 16S10 (Secondary).

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The path space of a higher-rank graph

Cited by 13 publications

References 10 publications

Kumjian-Pask algebras of desourcification

Kumjian-Pask algebras of desourcification

Primitive ideal space of higher-rank graph C⁎-algebras and decomposability

Cohn Path Algebras of Higher-Rank Graphs

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