Expanders, exact crossed products, and the Baum–Connes conjecture

Baum, Paul; Guentner, Erik; Willett, Rufus

doi:10.2140/akt.2016.1.155

Cited by 65 publications

(140 citation statements)

References 53 publications

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“…Furthermore, the existence of a fibred coarse embedding [6] implies the existence of an asymptotic embedding [3], so, in particular, we obtain the following. …”

Section: Coarse Properties Of the Cone Imply Equivariant Properties Omentioning

confidence: 71%

“…A reader familiar with the notion of asymptotic embedding [19] (asymptotically conditionally negative definite kernel of [3]), can immediately deduce from the proof that the assumptions of Proposition 6.1 (2) can be relaxed. Furthermore, the existence of a fibred coarse embedding [6] implies the existence of an asymptotic embedding [3], so, in particular, we obtain the following.…”

Section: Coarse Properties Of the Cone Imply Equivariant Properties Omentioning

confidence: 99%

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Warped cones over profinite completions

Sawicki

2018

J. Topol. Anal.

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Abstract. We construct metric spaces that do not have property A yet are coarsely embeddable into the Hilbert space. Our examples are so called warped cones, which were introduced by J. Roe to serve as examples of spaces nonembeddable into a Hilbert space and with or without property A. The construction provides the first examples of warped cones combining coarse embeddability and lack of property A.We also construct warped cones over manifolds with isometrically embedded expanders and generalise Roe's criteria for the lack of property A or coarse embeddability of a warped cone. Along the way, it is proven that property A of the warped cone over a profinite completion is equivalent to amenability of the group.In the appendix we solve a problem of Nowak regarding his examples of spaces with similar properties.

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“…Furthermore, the existence of a fibred coarse embedding [6] implies the existence of an asymptotic embedding [3], so, in particular, we obtain the following. …”

Section: Coarse Properties Of the Cone Imply Equivariant Properties Omentioning

confidence: 71%

Section: Coarse Properties Of the Cone Imply Equivariant Properties Omentioning

confidence: 99%

Warped cones over profinite completions

Sawicki

2018

J. Topol. Anal.

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“…It is somewhat frustrating that we do not know of any examples of exact large ideals other than B(G) (and, when G is exact, B r (G)). Perhaps other examples could be found using techniques similar to those of [BGW,Section 5].…”

Section: Resultsmentioning

confidence: 99%

Exact large ideals of 𝐵(𝐺) are downward directed

Kaliszewski

Landstad

Quigg

2016

Proc. Amer. Math. Soc.

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“…This nonexactness is responsible for counterexamples to the Baum-Connes conjecture in [56]. See [13,20,21] for a possible solution to this lack of exactness.…”

Section: Norm and C * -Algebramentioning

confidence: 99%

Lie groupoids, pseudodifferential calculus, and index theory

Debord¹,

Skandalis²

2019

Advances in Noncommutative Geometry

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Alain Connes introduced the use of Lie groupoids in noncommutative geometry in his pioneering work on the index theory of foliations. In the present paper, we recall the basic notion involved: groupoids, their C * -algebras, their pseudodifferential calculus... We review several recent and older advances on the involvement of Lie groupoids in noncommutative geometry. We then propose some open questions and possible developments of the subject. Lie groupoids and their operators algebrasWe refer to [78,80] for the classical definitions and constructions related to groupoids and their Lie algebroids. The construction of the C * -algebra of a groupoid is due to Jean Renault [100], one can look at his course [101] which is mainly devoted to locally compact groupoids. Lie groupoids GeneralitiesA groupoid is a small category in which every morphism is an isomorphism. Thus a groupoid G is a pair (G (0) , G (1) ) of sets together with structural morphisms:Units and arrows. The set G (0) denotes the set of objects (or units) of the groupoid, whereas the set G (1) is the set of morphisms (or arrows). The unit map u : G (0) → G (1) is the injective map which assigns to any object of G its identity morphism.The source and range maps s, r : G (1) → G (0) are (surjective) maps equal to identity in restriction toThe inverse ι : G (1) → G (1) is an involutive map which exchanges the source and range:for α ∈ G, (α −1 ) −1 = α and s(α −1 ) = r(α), where α −1 denotes ι(α)

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Expanders, exact crossed products, and the Baum–Connes conjecture

Cited by 65 publications

References 53 publications

Warped cones over profinite completions

Warped cones over profinite completions

Exact large ideals of 𝐵(𝐺) are downward directed

Lie groupoids, pseudodifferential calculus, and index theory

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