Remarks on some fundamental results about higher-rank graphs and their <i>C</i>*-algebras

Hazlewood, Robert; Raeburn, Iain; Sims, Aidan; Webster, Samuel B. G.

doi:10.1017/s0013091512000338

Cited by 50 publications

(70 citation statements)

References 24 publications

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“…The result now follows as in the proof of [10,Proposition 4.9] after observing that while [20,Lemma 6.2] is stated for aperiodic k-graphs the proof only requires the existence of an aperiodic path at the vertex v.…”

Section: Preliminaries On K-graphs and Kumjian-pask Algebrasmentioning

confidence: 78%

The Socle and Semisimplicity of a Kumjian–Pask Algebra

Brown

Huef

2015

Communications in Algebra

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Abstract. The Kumjian-Pask algebra KP(Λ) is a graded algebra associated to a higher-rank graph Λ and is a generalization of the Leavitt path algebra of a directed graph. We analyze the minimal left-ideals of KP(Λ), and identify its socle as a graded ideal by describing its generators in terms of a subset of vertices of the graph. We characterize when KP(Λ) is semisimple, and obtain a complete structure theorem for a semisimple Kumjian-Pask algebra.

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Section: Preliminaries On K-graphs and Kumjian-pask Algebrasmentioning

confidence: 78%

The Socle and Semisimplicity of a Kumjian–Pask Algebra

Brown

Huef

2015

Communications in Algebra

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“…We describe our examples using the 2-coloured graphs of [12]. A 2-coloured graph is a directed graph endowed with a map c : E 1 → {c 1 , c 2 }.…”

Section: Examplesmentioning

confidence: 99%

“…We think of c as determining a colour map from E * to the free abelian semigroup F 2 generated by {c 1 , c 2 } and for w ∈ F 2 , we say that λ ∈ E * is w-coloured if c(λ) = w. A collection of factorisation rules for E is a range-and source-preserving bijection θ from the c 1 c 2 -coloured paths in E * to the c 2 c 1 -coloured paths. For k = 2 the associativity condition of [12] is trivial, and so [12, Theorems 4.4 and 4.5] say that for every 2-coloured graph (E, c) with a collection θ of factorisation rules, there is a unique 2-graph Λ with Λ e i = c −1 (c i ), Λ 0 = E 0 , and ef = f ′ e ′ in Λ whenever θ(ef ) = f ′ e ′ in E * .…”

Section: Examplesmentioning

confidence: 99%

Real rank and topological dimension of higher rank graph algebras

Pask¹,

Sierakowski²,

Sims³

2017

Indiana Univ. Math. J.

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Abstract. We study dimension theory for the C * -algebras of row-finite k-graphs with no sources. We establish that strong aperiodicity-the higher-rank analogue of condition (K)-for a k-graph is necessary and sufficient for the associated C * -algebra to have topological dimension zero. We prove that a purely infinite 2-graph algebra has real-rank zero if and only if it has topological dimension zero and satisfies a homological condition that can be characterised in terms of the adjacency matrices of the 2-graph. We also show that a k-graph C * -algebra with topological dimension zero is purely infinite if and only if all the vertex projections are properly infinite. We show by example that there are strongly purely infinite 2-graphs algebras, both with and without topological dimension zero, that fail to have real-rank zero.

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“…One way to visualise k-graphs is to use coloured directed graphs, as described in [20]. Suppose Λ is a k-graph.…”

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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Remarks on some fundamental results about higher-rank graphs and their C*-algebras

Cited by 50 publications

References 24 publications

The Socle and Semisimplicity of a Kumjian–Pask Algebra

The Socle and Semisimplicity of a Kumjian–Pask Algebra

Real rank and topological dimension of higher rank graph algebras

Cohn Path Algebras of Higher-Rank Graphs

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