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2003 | Buch

Proper Group Actions and the Baum-Connes Conjecture

verfasst von: Guido Mislin, Alain Valette

Verlag: Birkhäuser Basel

Buchreihe : Advanced Courses in Mathematics - CRM Barcelona

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SUCHEN

Inhaltsverzeichnis

Frontmatter
Equivariant K-Homology of the Classifying Space for Proper Actions
Abstract
These notes are a compendium to a series of lectures concerning the topological aspects of the Baum-Connes Conjecture — the left hand side of the equation K * G (E G) ≅ K * top (C r * (G)) — the equivariant K-homology of E G. Besides of a presentation of the material needed to compute * G (E G,the reader will find an extensive discussion of many conjectures related to the Baum-Connes Conjecture.
Guido Mislin
On the Baum-Connes Assembly Map for Discrete Groups
Abstract
In these notes, we study the Baum-Connes analytical assembly maps (or index maps) μ i Γ : RK i Γ (EΓ) → K i (C r * Γ) and \( \bar \mu _i^\Gamma \) ,for a countable group Γ. Here RK i Γ denotes the Γ-equivariant K-homology with Γ-compact supports of the universal space E Γ for proper Γ-actions, while K i (C r * ) (resp. K i (C*Γ) denotes the analytical K-theory of the reduced (resp. full) C*-algebra of Γ. As it is simple and direct, we use the definition of β i Γ suggested by Baum, Connes and Higson in Section 3 of [BCH94]. The Baum-Connes conjecture asserts that, for any group Γ, the map β i Γ is an isomorphism (i= 0, 1). The contents of this paper are as follows:
1
We make the necessary changes for constructing \( \bar \mu _i^\Gamma \) , and give a detailed proof that β i Γ and \( \bar \mu _i^\Gamma \) provide K-theory elements of the corresponding C*-algebras.
 
2
We carefully describe the behavior of the left-hand side of the assembly maps under group homomorphisms, and we prove that \( \bar \mu _i^\Gamma \)is natural with respect to arbitrary group homomorphisms. As a consequence, we get a new proof of the fact that, if Γ acts freely on the space X, then the equibvalent K-homology K * Γ (X) is isomorphic to the H-homology K *(Γ\X) of the orbit space.
 
3
To illustrate the non-triviality of the assembly map, we give a direct proof of the Bauam-Connes conjecture for the group ℤ of integers, not appealing to equivariant KK-theory.
 
4
Denote by \( \tilde \kappa _\Gamma :\Gamma \to K_1 \left( {C_r^* \Gamma } \right) \) the homomorphism induced by the canonical inclusion of Γ in the unitary group of C r Γ . We show that there exixts a homomorphism \( \bar \beta _t :\Gamma \to RK_1^\Gamma \left( {\underline E T} \right) \) such that \( \bar \kappa _\Gamma = \mu _i^\Gamma \circ \tilde \beta _t \); this extends a result of Natsume [Nat88] for Γ torsion-free.
 
Alain Valette
Backmatter
Metadaten
Titel
Proper Group Actions and the Baum-Connes Conjecture
verfasst von
Guido Mislin
Alain Valette
Copyright-Jahr
2003
Verlag
Birkhäuser Basel
Electronic ISBN
978-3-0348-8089-3
Print ISBN
978-3-7643-0408-9
DOI
https://doi.org/10.1007/978-3-0348-8089-3