Non-Amenable Principal Groupoids with Weak Containment

Alekseev, V. G.; Finn-Sell, Martin

doi:10.1093/imrn/rnw305

Cited by 4 publications

(24 citation statements)

References 6 publications

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“…Willett [16] has constructed the first examples of (neither minimal nor essentially principal) non-amenableétale groupoids whose full and reduced groupoid C * -algebras coincide. Examples of principal groupoids with these conditions have been constructed by Alekseev-Finn-Sell [1]. Brodzki-Cave-Li [5] have established conjectured characterizations of exactness for locally compact groups (cf.…”

Section: Proposition C Let G Be a Locally Compact Non-amenable Exactmentioning

confidence: 99%

Simple equivariant C*-algebras whose full and reduced crossed products coincide

Suzuki

2020

J. Noncommut. Geom.

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For any second countable locally compact group G, we construct a simple G-C *algebra whose full and reduced crossed product norms coincide. We then construct its Gequivariant representation on another simple G-C * -algebra without the coincidence condition. This settles two problems posed by Anantharaman-Delaroche in 2002. Some constructions involve the Baire category theorem.2000 Mathematics Subject Classification. Primary 46L55, Secondary 46L05, 54H20.

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Section: Proposition C Let G Be a Locally Compact Non-amenable Exactmentioning

confidence: 99%

Simple equivariant C*-algebras whose full and reduced crossed products coincide

Suzuki

2020

J. Noncommut. Geom.

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“…So pu ˚pu, φpu ˚pq, pq « p0, 0, 0q and similarly pq, φpqwq, w ˚qwq « p0, 0, 0q. Now apply (1) The result then follows from part (3).…”

Section: Remark 43 (A General Setup) Let φmentioning

confidence: 97%

“…Recall first that if p : Y Ñ X and q : Z Ñ X are continuous maps, the fibered product is the space (1) Y p ˆq Z :" tpy, zq : ppyq " qpzqu.…”

Section: This Naturally Defines the Coinvariant Functormentioning

confidence: 99%

“…In particular, if G is a second countable, principal, minimal ample groupoid with finite dynamic asymptotic dimension, then C r pGq is classifiable. 1 Our result allows us to completely determine the classifying invariant (usually called the Elliott invariant) in the 1-dimensional case:…”

Section: Introductionmentioning

confidence: 97%

“…To see that this cannot be the case, we construct a counterexample using groupoids with topological property (T) introduced in [13]. The example is based on the construction of counterexamples to the Baum-Connes conjecture by Higson, Lafforgue and Skandalis [22] and work of Alekseev and Finn-Sell [1].…”

Section: Introductionmentioning

confidence: 99%

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Dynamic asymptotic dimension and Matui's HK conjecture

Bönicke¹,

Dell'Aiera²,

Gabe³

et al. 2021

Preprint

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We prove that the homology groups of a principal ample groupoid vanish in dimensions greater than the dynamic asymptotic dimension of the groupoid. As a consequence, the K-theory of the C ˚-algebras associated with groupoids of finite dynamic asymptotic dimension can be computed from the homology of the underlying groupoid. In particular, principal ample groupoids with dynamic asymptotic dimension at most two satisfy Matui's HK-conjecture.We also construct explicit maps from the groupoid homology groups to the K-theory groups of their C ˚-algebras in degrees zero and one, and investigate their properties. Contents 1. Introduction Outline of the paper 2. Models for groupoid homology 2.1. The category of G-modules and the coinvariant functor 2.2. Semi-simplicial G-spaces and homology 2.3. Projective resolutions and Tor 3. Colourings and homology 3.1. Colourings and nerves 3.2. Homology vanishing 3.3. Maps between nerves and G 3.4. Anti-Čech homology 3.5. Dynamic asymptotic dimension 4. The HK conjecture 4.1. The one-dimensional comparison map 4.2. Dynamic asymptotic dimension one 4.3. The spectral sequence of Proietti and Yamashita 5. Examples and applications 5.1. Free actions on totally disconnected spaces 5.2. Smale spaces with totally disconnected (un)stable sets 5.3. Bounded geometry metric spaces 5.4. Examples with topological property (T) References

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(Non)exotic Completions of the Group Algebras of Isotropy Groups

Christensen

Neshveyev

2021

International Mathematics Research Notices

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Motivated by the problem of characterizing KMS states on the reduced C$^*$-algebras of étale groupoids, we show that the reduced norm on these algebras induces a C$^*$-norm on the group algebras of the isotropy groups. This C$^*$-norm coincides with the reduced norm for the transformation groupoids, but, as follows from examples of Higson–Lafforgue–Skandalis, it can be exotic already for groupoids of germs associated with group actions. We show that the norm is still the reduced one for some classes of graded groupoids, in particular, for the groupoids associated with partial actions of groups and the semidirect products of exact groups and groupoids with amenable isotropy groups.

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Non-Amenable Principal Groupoids with Weak Containment

Cited by 4 publications

References 6 publications

Simple equivariant C*-algebras whose full and reduced crossed products coincide

Simple equivariant C*-algebras whose full and reduced crossed products coincide

Dynamic asymptotic dimension and Matui's HK conjecture

(Non)exotic Completions of the Group Algebras of Isotropy Groups

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