A dichotomy for groupoid -algebras

Rainone, Timothy; Sims, Aidan

doi:10.1017/etds.2018.52

Cited by 20 publications

(38 citation statements)

References 49 publications

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“…10 ([2, Theorem 5.1],[27, Corollary 6.6 and Theorem 7.4]). Let Γ be an ample groupoid such that Γ 0 is not compact and C * (Γ) is simple.…”

mentioning

confidence: 99%

KMS states of self-similar k-graph C*-algebras

Yang

2019

Journal of Functional Analysis

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Let (G, Λ) be a self-similar k-graph with a possibly infinite vertex set Λ 0 . We associate a universal C*-algebra OG,Λ to (G, Λ). The main purpose of this paper is to investigate the ideal structures of OG,Λ. We prove that there exists a one-to-one correspondence between the set of all G-hereditary and G-saturated subsets of Λ 0 and the set of all gauge-invariant and diagonal-invariant ideals of OG,Λ. Under some conditions, we characterize all primitive ideas of OG,Λ. Moreover, we describe the Jacobson topology of some concrete examples, which includes the C*-algebra of the product of odometers. On the way to our main results, we study self-similar P -graph C*-algebras in depth.2010 Mathematics Subject Classification. 46L05.

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“…10 ([2, Theorem 5.1],[27, Corollary 6.6 and Theorem 7.4]). Let Γ be an ample groupoid such that Γ 0 is not compact and C * (Γ) is simple.…”

mentioning

confidence: 99%

KMS states of self-similar k-graph C*-algebras

Yang

2019

Journal of Functional Analysis

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“…Now, we will show that Typ(G) coincides with some recently defined type semigroups [12,33,34]. To this end, we recall the definition in [12] (which is equivalent to the ones given in [33] and [34], as shown in [34]).…”

Section: Amenability Of G Tight (S(e C))mentioning

confidence: 92%

The Groupoids of Adaptable Separated Graphs and Their Type Semigroups

Ara

Bosa

Pardo

et al. 2020

International Mathematics Research Notices

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Given an adaptable separated graph, we construct an associated groupoid and explore its type semigroup. Specifically, we first attach to each adaptable separated graph a corresponding semigroup, which we prove is an E * -unitary inverse semigroup. As a consequence, the tight groupoid of this semigroup is a Hausdorffétale groupoid. We show that this groupoid is always amenable, and that the type semigroups of groupoids obtained from adaptable separated graphs in this way include all finitely generated conical refinement monoids. The first three named authors will utilize this construction in forthcoming work to solve the Realization Problem for von Neumann regular rings, in the finitely generated case.

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“…Also, [7,Corollary 4.9] gives a sufficient condition for an inner exact groupoid to have a purely infinite C * -algebra. [29,Proposition 7.1] gives a necessary condition on G for the C * -algebra to be purely infinite. How either of these conditions relates to a locally contracting groupoid is not immediately clear.…”

Section: The Edinburgh Mathematical Societymentioning

confidence: 99%

Dense subalgebras of purely infinite simple groupoidC*-algebras

Brown¹,

Clark²,

Huef³

2020

Proceedings of the Edinburgh Mathematical Society

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A simple Steinberg algebra associated to an ample Hausdorff groupoid G is algebraically purely infinite if and only if the characteristic functions of compact open subsets of the unit space are infinite idempotents. If a simple Steinberg algebra is algebraically purely infinite, then the reduced groupoid C * -algebra C * r (G) is simple and purely infinite. But the Steinberg algebra seems to small for the converse to hold. For this purpose we introduce an intermediate * -algebra B(G) constructed using corners 1 U C * r (G)1 U for all compact open subsets U of the unit space of the groupoid. We then show that if G is minimal and effective, then B(G) is algebraically properly infinite if and only if C * r (G) is purely infinite simple. We apply our results to the algebras of higher-rank graphs.

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A dichotomy for groupoid -algebras

Cited by 20 publications

References 49 publications

KMS states of self-similar k-graph C*-algebras

KMS states of self-similar k-graph C*-algebras

The Groupoids of Adaptable Separated Graphs and Their Type Semigroups

Dense subalgebras of purely infinite simple groupoidC*-algebras

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