The primitive ideals of the Cuntz–Krieger algebra of a row-finite higher-rank graph with no sources

Carlsen, Toke Meier; Kang, Sooran; Shotwell, Jacob; Sims, Aidan

doi:10.1016/j.jfa.2013.08.029

Cited by 40 publications

(122 citation statements)

References 13 publications

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“…For our purposes, it suffices to recall first that C * (Λ) is isomorphic to C * (G Λ ) [5,Proposition 2.7], and second that this isomorphism intertwines the gauge action of and (x, g, y) ∈ G Λ . An ideal of C * (Λ) is gauge-invariant if it is invariant for this gauge action.…”

Section: Applications To K-graphsmentioning

confidence: 99%

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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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Section: Applications To K-graphsmentioning

confidence: 99%

“…Choose any a ∈ C c (G Λ ) whose restriction to G Λ | [x] is equal to 1 U − χ(p)1 K . As in the proof of [5,Proposition 5.5], let π x be the representation of…”

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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“…Following [6] we recall the definition of Per(Λ) and H Per . Let Λ be a row-finite k-graph with no sources such that Λ 0 is a maximal tail.…”

Section: Topological Dimension Zeromentioning

confidence: 99%

“…Then [6, Lemma 4.2(1)] says that the set Per(Λ) := {d(µ) − d(ν) : µ, ν ∈ Λ and µ ∼ ν} is a subgroup of Z k , called the periodicity group of Λ. Moreover, it follows from [6,Lemma 4.6], that Per(Λ) = {l ∈ Z k : there exists w ∈ Λ 0 and n, m ∈ N k such that l = n − m and…”

Section: Topological Dimension Zeromentioning

confidence: 99%

“…We prove in Section 3 that this sufficient condition is also necessary (Theorem 3.2). Our approach uses the fact that topological dimension zero passes to ideals and quotients, together with recent results on P -graphs, induced algebras, and primitive-ideal structure of k-graph algebras [6,39]. We finish the section with Corollary 3.9 which describes a number of properties of Λ and C * (Λ) that are all equivalent to strong aperiodicity of Λ, and hence topological dimension zero for C * (Λ).…”

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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On Hong and Szymański’s Description of the Primitive-Ideal Space of a Graph Algebra

Carlsen¹,

Sims²

2016

Operator Algebras and Applications

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The primitive ideals of the Cuntz–Krieger algebra of a row-finite higher-rank graph with no sources

Cited by 40 publications

References 13 publications

A stabilization theorem for Fell bundles over groupoids

A stabilization theorem for Fell bundles over groupoids

Real rank and topological dimension of higher rank graph algebras

On Hong and Szymański’s Description of the Primitive-Ideal Space of a Graph Algebra

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