Simplicity of inverse semigroup and étale groupoid algebras

Steinberg, Benjamin; Szakács, Nóra

doi:10.1016/j.aim.2021.107611

Cited by 7 publications

(5 citation statements)

References 53 publications

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“…Secondly, pseudoexpectations are always supportive. This generalises Theorem 1 to all aperiodic inclusions A Ď B. Steinberg and Szakács in [59] prove a criterion when the Steinberg algebra of an étale groupoid with totally disconnected object space is simple. Our results imply an analogous criterion for reduced groupoid C ˚-algebras.…”

Section: Introductionmentioning

confidence: 53%

Essential crossed products for inverse semigroup actions: simplicity and pure infiniteness

Kwaśniewski

Meyer

2021

Doc. Math.

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We study simplicity and pure infiniteness criteria for C ˚algebras associated to inverse semigroup actions by Hilbert bimodules and to Fell bundles over étale not necessarily Hausdorff groupoids. Inspired by recent work of Exel and Pitts, we introduce essential crossed products for which there are such criteria. In our approach the major role is played by a generalised expectation with values in the local multiplier algebra. We give a long list of equivalent conditions characterising when the essential and reduced C ˚-algebras coincide. Our most general simplicity and pure infiniteness criteria apply to aperiodic C ˚-inclusions equipped with supportive generalised expectations. We thoroughly discuss the relationship between aperiodicity, detection of ideals, purely outer inverse semigroup actions, and non-triviality conditions for dual groupoids.

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Section: Introductionmentioning

confidence: 53%

Essential crossed products for inverse semigroup actions: simplicity and pure infiniteness

Kwaśniewski

Meyer

2021

Doc. Math.

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“…Hence for second-countable Hausdor↵ étale groupoids, all three notions agree (see also [52,Proposition 3.6]). However, there are examples of ample groupoids that are e↵ective but not topologically principal (see [10,Example 6.4], or [59,Section 6.2] for examples that are minimal, in the sense that r(G x ) is dense in G (0) for every x 2 G (0) ), and examples of second-countable non-Hausdor↵ groupoids that are étale and topologically principal but not e↵ective (see [16,Section 5.1]). The construction in [59,Proposition 6.11], but with the finite alphabet X replaced by an infinite one, yields a topologically free minimal ample groupoid that is neither e↵ective (by [59, Proposition 6.1]) nor topologically principal.…”

Section: 2mentioning

confidence: 99%

Reconstruction of Twisted Steinberg Algebras

Armstrong¹,

Castro

Clark

et al. 2021

International Mathematics Research Notices

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We show how to recover a discrete twist over an ample Hausdorff groupoid from a pair consisting of an algebra and what we call a quasi-Cartan subalgebra. We identify precisely which twists arise in this way (namely, those that satisfy the local bisection hypothesis), and we prove that the assignment of twisted Steinberg algebras to such twists and our construction of a twist from a quasi-Cartan pair are mutually inverse. We identify the algebraic pairs that correspond to effective groupoids and to principal groupoids. We also indicate the scope of our results by identifying large classes of twists for which the local bisection hypothesis holds automatically.

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“…compact open subsets (this was first proved in [27] for the Hausdorff case and in an unpublished note by Buss and Meyers in the general case; see [31] for a proof).…”

Section: Groupoidsmentioning

confidence: 83%

Stable finiteness of ample groupoid algebras, traces and applications

Steinberg¹

2022

Preprint

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In this paper we study stable finiteness of ample groupoid algebras with applications to inverse semigroup algebras and Leavitt path algebras, recovering old results and proving some new ones. In addition, we develop a theory of (faithful) traces on ample groupoid algebras, mimicking the C * -algebra theory but taking advantage of the fact that our functions are simple and so do not have integrability issues, even in the non-Hausdorff setting. The theory of traces is closely connected with the theory of invariant means on Boolean inverse semigroups. We include an appendix on stable finiteness of more general semigroup algebras, improving on an earlier result of Munn, which is independent of the rest of the paper.

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Simplicity of inverse semigroup and étale groupoid algebras

Cited by 7 publications

References 53 publications

Essential crossed products for inverse semigroup actions: simplicity and pure infiniteness

Essential crossed products for inverse semigroup actions: simplicity and pure infiniteness

Reconstruction of Twisted Steinberg Algebras

Stable finiteness of ample groupoid algebras, traces and applications

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