La conjecture de Baum–Connes pour les feuilletages moyennables

Tu, Jean-Louis

doi:10.1023/a:1007744304422

Cited by 250 publications

(259 citation statements)

References 30 publications

Supporting

Mentioning

251

Contrasting

Unclassified

Order By: Relevance

“…The proof of (ii) is a consequence of (i) and the fact that the strong BaumConnes conjecture implies the Baum-Connes conjecture for all G-algebras and it implies also that G is K-amenable which implies that the regular representation L : A ⋊ G → A ⋊ r G induces an isomorphism in K-theory [22].…”

Section: Examplesmentioning

confidence: 86%

Fibrations with noncommutative fibers

Echterhoff¹,

Nest²,

Oyono‐Oyono³

2009

J. Noncommut. Geom.

View full text Add to dashboard Cite

Abstract. We study an analogue of fibrations of topological spaces with the homotopy lifting property in the setting of C * -algebra bundles. We then derive an analogue of the Leray-Serre spectral sequence to compute the K-theory of the fibration in terms of the cohomology of the base and the K-theory of the fibres. We present many examples which show that fibrations with noncommutative fibres appear in abundance in nature. IntroductionIn recent years the study of the topological properties of C*-algebra bundles plays a more and more prominent rôle in the field of Operator algebras. The main reason for this is two-fold: on one side there are many important examples of C*-algebras which do come with a canonical bundle structure. On the other side, the study of C*-algebra bundles over a locally compact Hausdorff base space X is the natural next step in classification theory, after the far reaching results which have been obtained in the classification of simple C*-algebras. To fix notation, by a C*-algebra bundle A(X) over X we shall simply mean a C 0 (X)-algebra in the sense of Kasparov (see [17]): it is a C*-algebra A together with a non-degenerate * -homomorphismwhere ZM(A) denotes the center of the multiplier algebra M(A) of A. For such C 0 (X)-algebra A, the fibre over x ∈ X is then A x = A/I x , where I x = {Φ(f ) · a; a ∈ A and f ∈ C 0 (X) such that f (x) = 0}, and the canonical quotient map q x : A → A x is called the evaluation map at x. We shall often write A(X) to indicate the given C 0 (X)-structure of A. We shall recall the basic constructions and properties of C 0 (X)-algebras in the preliminary section below. We refer to [8] for further notations concerning C 0 (X)-algebras.The main problem when studying bundles from the topological point of view is to provide good topological invariants which help to understand the local and global structure of the bundles. A good example is given by the class of separable continuous-trace C*-algebras, which are, up to Morita equivalence, just the section algebras of locally trivial bundles over X with fibres the compact This work was partially supported by the Deutsche Forschungsgemeinschaft (SFB 478).

show abstract

Section: Examplesmentioning

confidence: 86%

Fibrations with noncommutative fibers

Echterhoff¹,

Nest²,

Oyono‐Oyono³

2009

J. Noncommut. Geom.

View full text Add to dashboard Cite

show abstract

“…Tu was also able to prove a groupoid version of the above mentioned result of Higson and Kasparov (see [21]). In particular, all topologically amenable groupoids (in the sense of [1]) satisfy BC for arbitrary coefficients.…”

mentioning

confidence: 82%

Shapiro's lemma for topological K-theory of groups

Chabert¹,

Echterhoff²,

Oyono‐Oyono³

2003

Commun Math Helv

View full text Add to dashboard Cite

Abstract. Let X G be the crossed product groupoid of a locally compact group G acting on a locally compact space X. For any X G-algebra A we show that a natural forgetful map from the topological K-theory K top * (X G; A) of the groupoid X G with coefficients in A to the topological K-theory K top * (G; A) of G with coefficients in A is an isomorphism. We then discuss several interesting consequences of this result for the Baum-Connes conjecture.

show abstract

“…This implies that the groupoid βΓ Γ is K-amenable [11]. Hence, we have that the C * -algebras C(βΓ) r Γ and C(βΓ) Γ are K-nuclear [10], [Proposition 4.17]. Hence, we get that Γ is K-exact, by Lemma 3.6.…”

Section: Lemma 32 If a Cmentioning

confidence: 98%