Aperiodicity and primitive ideals of row-finite k-graphs

Kang, Sooran; Pask, David

doi:10.1142/s0129167x14500220

Cited by 31 publications

(45 citation statements)

References 37 publications

(71 reference statements)

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“…We also show that this is equivalent to asking that for any saturated hereditary H ⊆ Λ 0 , the vertex projections of C * (Λ \ ΛH) are all infinite (not, a priori, properly infinite). Using this, we prove that if a k-graph has an aperiodic quartet [16] at every vertex, then its C * -algebra is purely infinite (Proposition 5.1). Our result adds to the list (see [11,21,37]) of known sufficient conditions on a k-graph Λ for pure infiniteness of its C * -algebra.…”

Section: Introductionmentioning

confidence: 94%

“…We say that Λ is aperiodic (or satisfies the aperiodicity condition) if for every vertex v ∈ Λ 0 there exist an infinite path x ∈ vΛ ∞ such that σ m (x) = σ n (x) for all m = n ∈ N k , see [21,32,36]. If Λ \ ΛH is aperiodic for every hereditary saturated H Λ 0 , we say Λ is strongly aperiodic, see [16].…”

Section: Topological Dimension Zeromentioning

confidence: 99%

“…For example Bratteli and Elliott showed [1] that a separable C * -algebra A of type I is AF if and only if it has topological dimension zero. For more recent work on topological dimension see [3,5,16,28,30,31,35,38].…”

Section: Introductionmentioning

confidence: 99%

“…They proved in [16,Theorem 4.2] that for a row-finite k-graph Λ with no sources, the C * -algebra associated to Λ has topological dimension zero if Λ is strongly aperiodic. Strong aperiodicity is the higher-rank analogue of the well-known condition (K) for directed graphs (every vertex that is the basepoint of a first-return path is the basepoint of at least two such paths).…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Introductionmentioning

confidence: 94%

Section: Topological Dimension Zeromentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Real rank and topological dimension of higher rank graph algebras

Pask¹,

Sierakowski²,

Sims³

2017

Indiana Univ. Math. J.

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“…Following the standard definition for k-graphs [32] (see also [37]), we say that a subset H ⊆ Λ 0 of the vertex set of a row-finite P -graph Λ with no sources is hereditary if HΛ ⊆ ΛH and saturated if whenever vΛ p ⊆ ΛH, we have v ∈ H. A subset T of Λ 0 is a maximal tail if its complement is a saturated hereditary set and s(vΛ) ∩ s(wΛ) = ∅ for all v, w ∈ T . We say that Λ is strongly aperiodic if for every saturated hereditary subset H ⊆ Λ 0 , the subgraph Λ \ ΛH is aperiodic (see [19]). (2) =⇒ (3) If Λ \ ΛH is aperiodic for every saturated hereditary H, then in particular, every ΛT is aperiodic because the complement of a maximal tail is saturated and hereditary.…”

Section: Applications To K-graphsmentioning

confidence: 99%

A stabilization theorem for Fell bundles over groupoids

Ionescu¹,

Kumjian²,

Sims³

et al. 2017

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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ABSTRACT. We study the C * -algebras associated to upper-semicontinuous Fell bundles over second-countable Hausdorff groupoids. Based on ideas going back to the Packer-Raeburn "Stabilization Trick," we construct from each such bundle a groupoid dynamical system whose associated Fell bundle is equivalent to the original bundle. The upshot is that the full and reduced C * -algebras of any saturated upper-semicontinuous Fell bundle are stably isomorphic to the full and reduced crossed products of an associated dynamical system. We apply our results to describe the lattice of ideals of the C * -algebra of a continuous Fell bundle by applying Renault's results about the ideals of the C * -algebras of groupoid crossed products. In particular, we discuss simplicity of the Fell-bundle C * -algebra of a bundle over G in terms of an action, described by the first and last named authors, of G on the primitive-ideal space of the C * -algebra of the part of the bundle sitting over the unit space. We finish with some applications to twisted k-graph algebras, where the components of our results become more concrete.

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Wavelets and Graph C ∗-Algebras

Farsi

Gillaspy

Kang

et al. 2017

Excursions in Harmonic Analysis, Volume 5

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Here we give an overview on the connection between wavelet theory and representation theory for graph C * -algebras, including the higher-rank graph C * -algebras of A. Kumjian and D. Pask. Many authors have studied different aspects of this connection over the last 20 years, and we begin this paper with a survey of the known results. We then discuss several new ways to generalize these results and obtain wavelets associated to representations of higher-rank graphs. In [20], we introduced the "cubical wavelets" associated to a higher-rank graph. Here, we generalize this construction to build wavelets of arbitrary shapes. We also present a different but related construction of wavelets associated to a higher-rank graph, which we anticipate will have applications to traffic analysis on networks. Finally, we generalize the spectral graph wavelets of [22] to higher-rank graphs, giving a third family of wavelets associated to higher-rank graphs.2010 Mathematics Subject Classification: 46L05, 42C40

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Aperiodicity and primitive ideals of row-finite k-graphs

Cited by 31 publications

References 37 publications

Real rank and topological dimension of higher rank graph algebras

Real rank and topological dimension of higher rank graph algebras

A stabilization theorem for Fell bundles over groupoids

Wavelets and Graph C ∗-Algebras

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