Purely infinite -algebras associated to étale groupoids

Brown, Jonathan H.; Clark, Lisa Orloff; Sierakowski, Adam

doi:10.1017/etds.2014.47

Cited by 19 publications

(27 citation statements)

References 21 publications

(26 reference statements)

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“…This topic has been intensively studied, and for 1-graphs this culminated in [13], where sufficient and necessary conditions for pure infiniteness of a 1-graph C * -algebra (in terms of the 1-graph) where established. Building on recent results for cofinal k-graphs in [2], we establish a number of properties equivalent to pure infiniteness of C * (Λ). In particular, generalising [2, Corollary 5.1], we prove that C * (Λ) is purely infinite if and only if the vertex projections {s v : v ∈ Λ 0 } are all properly infinite (without assuming Λ is cofinal).…”

Section: Introductionmentioning

confidence: 94%

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%

Real rank and topological dimension of higher rank graph algebras

Pask¹,

Sierakowski²,

Sims³

2017

Indiana Univ. Math. J.

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“…Accordingly, this paper will be concerned with purely infinite C * -algebras associated to étale groupoids. One step in this direction has already been done in [9], where the authors focus on the simple setting. We extend the results of [9] in two directions: Firstly we drop the minimality condition, which leads us to the realm of non-simple C * -algebras.…”

Section: Introductionmentioning

confidence: 99%

“…One step in this direction has already been done in [9], where the authors focus on the simple setting. We extend the results of [9] in two directions: Firstly we drop the minimality condition, which leads us to the realm of non-simple C * -algebras. And secondly, we give a sufficient condition on the groupoid for pure infiniteness of its reduced groupoid C * -algebra.…”

Section: Introductionmentioning

confidence: 99%

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Ideal structure and pure infiniteness of ample groupoid -algebras

Bönicke¹,

Li²

2018

Ergod. Th. Dynam. Sys.

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In this paper, we study the ideal structure of reduced C * -algebras C * r (G) associated to étale groupoids G. In particular, we characterize when there is a one-to-one correspondence between the closed, two-sided ideals in C * r (G) and the open invariant subsets of the unit space G (0) of G. As a consequence, we show that if G is an inner exact, essentially principal, ample groupoid, then C * r (G) is (strongly) purely infinite if and only if every non-zero projection in C0(G (0) ) is properly infinite in C * r (G). We also establish a sufficient condition on the ample groupoid G that ensures pure infiniteness of C * r (G) in terms of paradoxicality of compact open subsets of the unit space G (0) .Finally, we introduce the type semigroup for ample groupoids and also obtain a dichotomy result: Let G be an ample groupoid with compact unit space which is minimal and topologically principal. If the type semigroup is almost unperforated, then C * r (G) is a simple C * -algebra which is either stably finite or strongly purely infinite.

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“…ii) ⇒ (iii): By the main result in[5] it suffices to check that every non-empty compact open set A ⊆ G (0) defines a properly infinite projection 1 A in C * r (G). By our assumption every such A is (2, 1)-paradoxical, and Lemma 7.2 implies that 1 A is properly infinite.…”

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confidence: 99%

A dichotomy for groupoid -algebras

Rainone

Sims

2018

Ergod. Th. Dynam. Sys.

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We study the finite versus infinite nature of C * -algebras arising frométale groupoids. For an ample groupoid G, we relate infiniteness of the reduced C * -algebra of G to notions of paradoxicality of a K-theoretic flavor. We construct a pre-ordered abelian monoid S(G) which generalizes the type semigroup introduced by Rørdam and Sierakowski for totally disconnected discrete transformation groups. This monoid reflects the finite/infinite nature of the reduced groupoid C * -algebra of G. If G is ample, minimal, and topologically principal, and S(G) is almost unperforated we obtain a dichotomy between stable finiteness and pure infiniteness for the reduced C * -algebra of G.

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Purely infinite -algebras associated to étale groupoids

Cited by 19 publications

References 21 publications

Real rank and topological dimension of higher rank graph algebras

Real rank and topological dimension of higher rank graph algebras

Ideal structure and pure infiniteness of ample groupoid -algebras

A dichotomy for groupoid -algebras

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