The Gamma Element for Groups which Admit a Uniform Embedding into Hilbert Space

Tu, Jean-Louis

doi:10.1007/3-7643-7314-8_18

Cited by 9 publications

(17 citation statements)

References 15 publications

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“…Let us recall that Hilbert space uniform embeddability of G implies that is split injective, as proven by Yu [29] if the classifying space BG is finite and by Skandalis, Yu and Tu [24] in the general case. We will also use a strengthening of this result by Tu [26] who showed that G has a gamma element. In conjunction with a theorem of Kasparov [15] this guarantees the surjectivity of the dual assembly map W K 0 .C .G// !…”

Section: Introductionmentioning

confidence: 92%

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Group quasi-representations and almost flat bundles

Dǎdǎrlat¹

2014

J. Noncommut. Geom.

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We study the existence of quasi-representations of discrete groups G into unitary groups U.n/ that induce prescribed partial maps K 0 .C .G// ! Z on the K-theory of the group C*-algebra of G. We give conditions for a discrete group G under which the K-theory group of the classifying space BG consists entirely of almost flat classes. (2010). 46L80, 46L65, 19K35, 19K56. Mathematics Subject Classification

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Section: Introductionmentioning

confidence: 92%

“…Z. We rely heavily on results of Kasparov, Higson, Yu, Skandalis and Tu [15], [12], [29], [24], [16], [26]. For illustration, we have the following: Theorem 1.1.…”

Section: Introductionmentioning

confidence: 99%

Group quasi-representations and almost flat bundles

Dǎdǎrlat¹

2014

J. Noncommut. Geom.

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“…where EΓ is the classifying space for proper actions of Γ and p : EΓ Γ → Γ is the homomorphism defined by p(z, g) = g. We refer the reader to [28,29]. The existence of the γ element implies that the Baum-Connes assembly map (with coefficients) is split injective and that the group is K-amenable: this last property gives the existence of a non trivial element ζ ∈ KK(C * r (Γ 2 ), C) such that, if ξ = [L 2 ( X 2 ), D] ∈ KK Γ2 ( X 2 , C) is the class given by an equivariant elliptic operator D, then µ Γ2 X2 (D) ⊗ C * r (Γ2) ζ is equal to the Fredholm index of the induced operator on X 2 /Γ.…”

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confidence: 99%

Mapping the surgery exact sequence for topological manifolds to analysis

Zenobi

2017

J. Topol. Anal.

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In this paper we prove the existence of a natural mapping from the surgery exact sequence for topological manifolds to the analytic surgery exact sequence of N. Higson and J. Roe. This generalizes the fundamental result of Higson and Roe, but in the treatment given by Piazza and Schick, from smooth manifolds to topological manifolds. Crucial to our treatment is the Lipschitz signature operator of Teleman.We also give a generalization to the equivariant setting of the product defined by Siegel in his Ph.D. thesis. Geometric applications are given to stability results for rho classes. We also obtain a proof of the APS delocalised index theorem on odd dimensional manifolds, both for the spin Dirac operator and the signature operator, thus extending to odd dimensions the results of Piazza and Schick. Consequently, we are able to discuss the mapping of the surgery sequence in all dimensions. Signature operator on Lipschitz manifoldsWe start recalling fundamental results on Lipschitz manifolds. For further details we refer to [26,27,8,25,30].Definition 2.1. A Lipschitz atlas on a topological manifold M is an atlas such that the map ϕ • ψ −1 is a Lipschitz homeomorphism for any two charts ϕ : U → R n and ψ : V → R n . By definition a Lipschitz manifold structure on M is a maximal Lipschitz atlas.

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“…since G has a Dirac element coming from a proper σ-G-C * -algebra when G is discrete (Theorem 2.1 of [Tu05]) or a closed subgroup of an almost connected second countable group H (Theorem 4.8 of [Kas88]).…”

Section: If G Satisfies (51) In Addition There Is An Open Coveringmentioning

confidence: 99%

A categorical perspective on the Atiyah–Segal completion theorem in KK-theory

Arano

Kubota

2018

J. Noncommut. Geom.

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We investigate the homological ideal J H G , the kernel of the restriction functors in compact Lie group equivariant Kasparov categories. Applying the relative homological algebra developed by Meyer and Nest, we relate the Atiyah-Segal completion theorem with the comparison of J H G with the augmentation ideal of the representation ring. In relation to it, we study on the Atiyah-Segal completion theorem for groupoid equivariant KK-theory, McClure's restriction map theorem and permanence property of the Baum-Connes conjecture under extensions of groups.

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The Gamma Element for Groups which Admit a Uniform Embedding into Hilbert Space

Cited by 9 publications

References 15 publications

Group quasi-representations and almost flat bundles

Group quasi-representations and almost flat bundles

Mapping the surgery exact sequence for topological manifolds to analysis

A categorical perspective on the Atiyah–Segal completion theorem in KK-theory

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