A uniqueness theorem for twisted groupoid C*-algebras

Armstrong, Becky

doi:10.1016/j.jfa.2022.109551

Cited by 5 publications

(12 citation statements)

References 31 publications

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“…We denote by E (0) the unit space of E. The range and source maps r, s : E → E (0) are given by r(e) = ee −1 and s(e) = e −1 e, respectively. The set of composable pairs {(d, e) : s(d) = r(e)} is denoted by E (2) . Then E is principal if the map e → (r(e), s(e)) is injective.…”

Section: Preliminariesmentioning

confidence: 99%

“…The existence of continuous local sections and the local trivializations are often assumed in the definition of a twist over an étale groupoid (see, for example, [39]), and it then follows that π is automatically an open map. Indeed, when G is étale, [39, Definition 11.1.1] and Definition 2.1 are equivalent, as outlined in [2,Remark 2.6].…”

Section: 2mentioning

confidence: 99%

“…is open in G and in E, and hence ι is an open map into E as it is a homeomorphism onto an open subset. If G is even étale, then the range and source maps on E are open, and the multiplication map on E (2) is open (see, for example, [2, Lemma 2.7]).…”

Section: 2mentioning

confidence: 99%

“…Let (E, ι, π) be a twist over a locally compact, Hausdorff, r-discrete groupoid G with non-compact unit space G (0) . Set E := E ∪ {∞ z : z ∈ T} and E (2)…”

Section: Unitizationsmentioning

confidence: 99%

“…The twists E that give rise to classifiable C * -algebras are over second-countable, locally compact, Hausdorff étale groupoids G which satisfy an aperiodicity condition called effectiveness, and the associated reduced twisted groupoid C * -algebra C * r (E; G) is simple if and only if G is minimal [2]. If C * r (E; G) is nuclear, then we know from [4,42] that it automatically satisfies the UCT.…”

Section: Introductionmentioning

confidence: 99%

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Alexandrov groupoids and the nuclear dimension of twisted groupoid $\mathrm{C}^*$-algebras

Courtney¹,

Duwenig²,

Huef³

et al. 2022

Preprint

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We consider a twist E over an étale groupoid G. When G is principal, we prove that the nuclear dimension of the reduced twisted groupoid C * -algebra is bounded by a number depending on the dynamic asymptotic dimension of G and the topological covering dimension of its unit space. This generalizes an analogous theorem by Guentner, Willett, and Yu for the C * -algebra of G. Our proof uses a reduction to the unital case where G has compact unit space, via a construction of "groupoid unitizations" G and E of G and E such that E is a twist over G. The construction of G is for r-discrete (hence étale) groupoids G which are not necessarily principal. When G is étale, the dynamic asymptotic dimension of G and G coincide. We show that the minimal unitizations of the full and reduced twisted groupoid C * -algebras of the twist over G are isomorphic to the twisted groupoid C * -algebras of the twist over G. We apply our result about the nuclear dimension of the twisted groupoid C * -algebra to obtain a similar bound on the nuclear dimension of the C * -algebra of an étale groupoid with closed orbits and abelian stability subgroups that vary continuously.

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Section: Preliminariesmentioning

confidence: 99%

Section: 2mentioning

confidence: 99%

Section: 2mentioning

confidence: 99%

“…Let (E, ι, π) be a twist over a locally compact, Hausdorff, r-discrete groupoid G with non-compact unit space G (0) . Set E := E ∪ {∞ z : z ∈ T} and E (2)…”

Section: Unitizationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Alexandrov groupoids and the nuclear dimension of twisted groupoid $\mathrm{C}^*$-algebras

Courtney¹,

Duwenig²,

Huef³

et al. 2022

Preprint

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The local bisection hypothesis for twisted groupoid C*-algebras

Armstrong,

Brown,

Clark

et al. 2023

Semigroup Forum

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In this note, we present criteria that are equivalent to a locally compact Hausdorff groupoid G being effective. One of these conditions is that G satisfies the C*-algebraic local bisection hypothesis; that is, that every normaliser in the reduced twisted groupoid C*-algebra is supported on an open bisection. The semigroup of normalisers plays a fundamental role in our proof, as does the semigroup of normalisers in cyclic group C*-algebras.

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