The Baum-Connes conjecture: an extended survey

Aparicio, Maria Paula Gomez; Julg, Pierre; Valette, Alain

doi:10.48550/arxiv.1905.10081

Cited by 4 publications

(5 citation statements)

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“…This conjecture has been proven for various classes of metric space and disproven for a few others, most notably expanders and warped cones [17,32]. The interested reader is referred to [2] and references therein for more information about the conjecture.…”

Section: Averaging Projectionsmentioning

confidence: 95%

“…• either P X does not belong to the uniform Roe algebra-in which case we just proved that C * u (X) C * uq (X); • or P X is in the uniform Roe algebra-in which case we are in a good position for providing further counterexamples to the coarse Baum-Connes conjecture. 2 One possible way to distinguish whether P X ∈ C * u (X) or not is based on a result by Finn-Sell [11], from which one can deduce that P X C * u (X) if {X n } n∈N uniformly coarsely embeds into some Hilbert space (Subsection 2.1). This raises the question [21,Question 7.3] of whether there exist asymptotic expanders with bounded geometry that can be coarsely embedded into Hilbert spaces.…”

Section: Introductionmentioning

confidence: 99%

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On the structure of asymptotic expanders

Khukhro

Vigolo

et al. 2021

Advances in Mathematics

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Section: Averaging Projectionsmentioning

confidence: 95%

Section: Introductionmentioning

confidence: 99%

On the structure of asymptotic expanders

Khukhro

Vigolo

et al. 2021

Advances in Mathematics

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“…This conjecture has been proven for various classes of metric space and disproven for a few others, most notably expanders and warped cones [16,28]. The interested reader is referred to [2] and references therein for more information about the conjecture.…”

Section: Roe Algebras and Quasi-local Algebrasmentioning

confidence: 95%

On the structure of asymptotic expanders

Khukhro¹,

Li²,

Vigolo³

et al. 2019

Preprint

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In this paper, we use geometric tools to study the structure of asymptotic expanders and show that a sequence of asymptotic expanders always admits a "uniform exhaustion by expanders". It follows that asymptotic expanders cannot be coarsely embedded into any L p -space, and that asymptotic expanders can be characterised in terms of their uniform Roe algebra. Moreover, we provide uncountably many new counterexamples to the coarse Baum-Connes conjecture. These appear to be the first counterexamples that are not directly constructed by means of spectral gaps. Finally, we show that vertex-transitive asymptotic expanders are actually expanders. In particular, this gives a C * -algebraic characterisation of expanders for vertex-transitive graphs.

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“…(2) (G) is irrational) with 0 isolated in the spectrum of ∆ n will not satisfy the Baum-Connes conjecture; more precisely, the Baum-Connes assembly map for G will not be surjective. We refer to [12,27] for an overview of the Baum-Connes conjecture. It is worth noting that such a counterexample, with bounded orders of finite subgroups, would also not satisfy the Atiyah conjecture, see e.g.…”

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confidence: 99%

Higher Kazhdan projections, $\ell_2$-Betti numbers and Baum-Connes conjectures

Li,

Nowak,

Pooya

2020

Preprint

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We introduce higher-dimensional analogs of Kazhdan projections in matrix algebras over group C * -algebras and Roe algebras. These projections are constructed in the framework of cohomology with coefficients in unitary representations and in certain cases give rise to non-trivial K-theory classes. We apply the higher Kazhdan projections to establish a relation between ℓ 2 -Betti numbers of a group and surjectivity of different Baum-Connes type assembly maps.Kazhdan projections are certain idempotents whose existence in the maximal group C * -algebra C * max (G) of a group G characterizes Kazhdan's property (T) for G. They have found an important application in higher index theory, as the non-zero K -theory class represented by a Kazhdan projection in K 0 (C * max (G)) obstructs the Dirac-dual Dirac method of proving the Baum-Connes conjecture. This idea has been used effectively in the coarse setting [15,16], where Kazhdan projections were used to construct counterexamples to the coarse Baum-Connes conjecture and to the Baum-Connes conjecture with coefficients. However, Kazhdan projections related to property (T) are not applicable to the classical Baum-Connes conjecture, as the image of a Kazhdan projection in the reduced group C * -algebra vanishes for any infinite group.The goal of this work is to introduce higher-dimensional analogs of Kazhdan projections with the motivation of applying them to the K -theory of group C * -algebra and to higher index theory. These higher Kazhdan projections are defined in the context of higher cohomology with coefficients in unitary representations and they are elements of matrix algebras over group C * -algebras. Their defining feature is that their image in certain unitary representations is the orthogonal projection onto the harmonic n-cochains. We show that the

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The Baum-Connes conjecture: an extended survey

Cited by 4 publications

References 0 publications

On the structure of asymptotic expanders

On the structure of asymptotic expanders

On the structure of asymptotic expanders

Higher Kazhdan projections, $\ell_2$-Betti numbers and Baum-Connes conjectures

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