Twisted Equivariant KK-Theory and the Baum–Connes Conjecture for Group Extensions

In this paper we study the stability of the Baum–Connes conjecture with coefficients undertaking group extensions. For this, it is necessary to extend Kasparov's equivariant KK-theory to an equivariant theory for twisted actions of groups on C ^*-algebras. As a consequence of our stability results, we are able to reduce the problem of whether closed subgroups of connected groups satisfy the Baum–Connes conjecture, with coefficients to the special case of center-free semi-simple Lie groups.     

Quantum Mechanics and Representation Theory: The New Synthesis

Quantum mechanics and representation theory, in the sense of unitary representations of groups on Hilbert spaces, were practically born together between 1925–1927, and have continued to enrich each other till the present day. Following a brief historical introduction, we focus on a relatively new aspect of the interaction between quantum mechanics and representation theory, based on the use of K-theory of C ^*-algebras. In particular, the study of the K-theory of the reduced C ^*-algebra of a locally compact group (which for a compact group is just its representation ring) has culminated in two fundamental conjectures, which are closely related to quantum theory and index theory, namely the Baum–Connes conjecture and the Guillemin–Sternberg conjecture. Although these conjectures were both formulated in 1982, and turn out to be closely related, so far there has been no interplay between them whatsoever, either mathematically or sociologically. This is presumably because the Baum–Connes conjecture is nontrivial only for noncompact groups, with current emphasis entirely on discrete groups, whereas the Guillemin–Sternberg conjecture has so far only been stated for compact Lie groups. As an elementary introduction to both conjectures in one go, indicating how the latter can be generalized to the noncompact case, this paper is a modest attempt to change this state of affairs.     

La conjecture de Baum–Connes pour les feuilletages moyennables

We show, using the construction of Higson and Kasparov, that the Baum–Connes Conjecture holds for foliations whose holonomy groupoid is Hausdorff and amenable. More generally, for every locally compact, -compact and Hausdorff groupoid G acting continuously and isometrically on a continuous field of affine Euclidean spaces, the Baum–Connes conjecture with coefficients is an isomorphism, and G amenable in K-theory. In addition, we show that C^*(G) satisfies the Universal Coefficient Theorem.     

Fibred coarse embeddings,a-T-menability and the coarse analogue of the Novikov conjecture

The connection between the coarse geometry of metric spaces and analytic properties of topological groupoids is well known. One of the main results of Skandalis, Tu and Yu is that a space admits a coarse embedding into Hilbert space if and only if a certain associated topological groupoid is a-T-menable. This groupoid characterisation then reduces the proof that the coarse Baum–Connes conjecture holds for a coarsely embeddable space to known results for a-T-menable groupoids. The property of admitting a fibred coarse embedding into Hilbert space was introduced by Chen, Wang and Yu to provide a property that is sufficient for the maximal analogue to the coarse Baum–Connes conjecture and in this paper we connect this property to the traditional coarse Baum–Connes conjecture using a restriction of the coarse groupoid and homological algebra. Additionally we use this results to give a characterisation of the a-T-menability for residually finite discrete groups.     

The maximal coarse Baum–Connes conjecture for spaces which admit a fibred coarse embedding into Hilbert space

We introduce a notion of fibred coarse embedding into Hilbert space for metric spaces, which is a generalization of Gromov?s notion of coarse embedding into Hilbert space. It turns out that a large class of expander graphs admit such an embedding. We show that the maximal coarse Baum–Connes conjecture holds for metric spaces with bounded geometry which admit a fibred coarse embedding into Hilbert space.     

Baum-Connes conjecture and coarse geometry

In this paper, we show that the Baum-Connes conjecture for a discrete group with coefficients inl (,K) is equivalent to the coarse Baum-Connes conjecture for as a metric space with a length metric. We apply this result to prove special cases of the Baum-Connes conjecture.Supported by DMS8505550 through a MSRI Postdoctoral Fellowship.     

Coronae of product spaces and the coarse Baum–Connes conjecture

We study the coarse Baum–Connes conjecture for product spaces and product groups. We show that a product of CAT(0) groups, polycyclic groups and relatively hyperbolic groups which satisfy some assumptions on peripheral subgroups, satisfies the coarse Baum–Connes conjecture. For this purpose, we construct and analyze an appropriate compactification and its boundary, “corona”, of a product of proper metric spaces.     

K-Theory for C*-Algebras of One-Relator Groups

We compute the K-theory groups of the reduced C*-algebra C _r ^* () of a one-relator group . We prove that every such group is K-amenable in the sense of Cuntz. For a torsion-free one-relator group =<X|r> such that r is not a product of commutators, we give a direct proof of the fact that the Baum–Connes analytical assembly map

Local-Global Principle for the Baum–Connes Conjecture with Coefficients

We establish the Hasse principle (local-global principle) in the context of the Baum–Connes conjecture with coefficients. We illustrate this principle with the discrete group GL(2,F) where F is any global field.     

Noncommutative simplicial complexes and the Baum—Connes conjecture

We associate a noncommutative C ^*-algebra with every locally finite simplicial complex.We determine the K-theory of these algebras and show that they can be used to obtain a conceptual explanation for the Baum—Connes conjecture. Our main technical result determines the algebra generated by the coefficients of a universal projection in an algebraic crossed product by a discrete group , in terms of such a noncommutative algebra associated with a simplicial complex defined by . Submitted: May 2001, Revised: October 2001, Revised: April 2002.     

K-Homology Relative to Semisplit Ideals

Given a C^*-algebra A, and an ideal J of A, we define a relative group K^⁰(A,J) in terms of a relative universal C^*-algebra for the pair (A,J). We show that the natural restriction map K^⁰(A,J) K⁰(J) is an isomorphism, and that, if J is a semisplit ideal of A, the Baum–Douglas–Taylor relative K-homology is recovered. This provides a generalization of one of the main results of the Baum–Douglas–Taylor theory.     

The Baum–Connes conjecture for free orthogonal quantum groups

We prove an analogue of the Baum–Connes conjecture for free orthogonal quantum groups. More precisely, we show that these quantum groups have a γ-element and that γ=1. It follows that free orthogonal quantum groups are K-amenable. We compute explicitly their K-theory and deduce in the unimodular case that the corresponding reduced C^?-algebras do not contain nontrivial idempotents.Our approach is based on the reformulation of the Baum–Connes conjecture by Meyer and Nest using the language of triangulated categories. An important ingredient is the theory of monoidal equivalence of compact quantum groups developed by Bichon, De Rijdt and Vaes. This allows us to study the problem in terms of the quantum group SUq(2). The crucial part of the argument is a detailed analysis of the equivariant Kasparov theory of the standard Podle? sphere.     

Baum–Connes Conjecture and Group Actions on Trees

We check, that the Baum–Connes conjecture with coefficients, for groups acting on oriented trees, is true if and only if it is true for the stabilizer groups of the vertices.     

Assembly maps,K-theory, and hyperbolic groups

Following Connes and Moscovici, we show that the Baum-Connes assembly map forK _*(C*v) is rationally injective when is word-hyperbolic, implying the Equivariant Novikov conjecture for such groups. Using this result in topologicalK-theory and Borel-Karoubi regulators, we also show that the corresponding generalized assembly map in algebraicK-theory is rationally injective.     

The Novikov Conjecture for Low Degree Cohomology Classes

We outline a twisted analogue of the Mishchenko–Kasparov approach to prove the Novikov conjecture on the homotopy invariance of the higher signatures. Using our approach, we give a new and simple proof of the homotopy invariance of the higher signatures associated to all cohomology classes of the classifying space that belong to the subring of the cohomology ring of the classifying space that is generated by cohomology classes of degree less than or equal to 2, a result that was first established by Connes and Gromov and Moscovici using other methods. A key new ingredient is the construction of a tautological C* _r (, )-bundle and connection, which can be used to construct a C* _r (, )-index that lies in the Grothendieck group of C* _r (, ), where is a multiplier on the discrete group corresponding to a degree 2 cohomology class. We also utilise a main result of Hilsum and Skandalis to establish our theorem.     

Fonctorialité en K-Théorie bivariante pour les variétés lipschitziennes

Résumé Soit f: M V/F un morphisme continu orienté d'une variété lipschitzienne M dans l'espace des feuilles d'une variété lipschitzienne feuilletée (V,F), et soit C ^* (V,F) la C ^*-algèbre du feuilletage d'A. Connes. On construit un élèment (f) dans le groupe de K-théorie bivariante KK(C ₀ (M); C ^* (V,F)) de G. G. Kasparov et on montre la fonctorialité de cette construction. On utilise l'opérateur de signature de N. Teleman ([42]). Ceci répond pour les variétés lipschitziennes à une conjecture d'A. Connes ([11]) qui a été résolue pour les variétés différentiables dans [13, 8, 19].

---

Let M be a Lipschitz manifold, (V, F) a foliated Lipschitz manifold and let f M V/F be an oriented morphism. Let C ^* (V,F) be the foliation's C ^*-algebra of A. Connes. We then construct an element (f) of the K-theory bivariant group KK(C ₀(M); C ^* (V, F)) of G. G. Kasparov which depends functoriality on f. This uses the signature operator of N. Teleman [42]. It gives a positive answer for Lipschitz manifolds to a conjecture of A. Connes [11] which has been proved for differentiable manifolds in [13, 8, 19].

     

Asymptotic Morphisms and Elliptic Operators over C*‐Algebras

This paper provides an Etheoretic proof of an exact form, due to E. Troitsky, of the Mischenko–Fomenko Index Theorem for elliptic pseudodifferential operators over a unital C^*algebra. The main ingredients in the proof are the use of asymptotic morphisms of Connes and Higson, vector bundle modification, a Baum–Douglastype group, and a K Kargument of Kasparov.     

Operator norm localization property of metric spaces under finite decomposition complexity

The notions of operator norm localization property and finite decomposition complexity were recently introduced in metric geometry to study the coarse Novikov conjecture and the stable Borel conjecture. In this paper we show that a metric space X has weak finite decomposition complexity with respect to the operator norm localization property if and only if X itself has the operator norm localization property. It follows that any metric space with finite decomposition complexity has the operator norm localization property. In particular, we obtain an alternative way to prove a very recent result by E. Guentner, R. Tessera and G. Yu that all countable linear groups have the operator norm localization property.     

On coarse geometric aspects of Hilbert geometry

We begin a coarse geometric study of Hilbert geometry. Actually we give a necessary and sufficient condition for the natural boundary of a Hilbert geometry to be a corona, which is a nice boundary in coarse geometry. In addition, we show that any Hilbert geometry is uniformly contractible and with coarse bounded geometry. As a consequence of these we see that the coarse Novikov conjecture holds for a Hilbert geometry with a mild condition. Also we show that the asymptotic dimension of any two-dimensional Hilbert geometry is just two. This implies that the coarse Baum–Connes conjecture holds for any two-dimensional Hilbert geometry via Yu’s theorem.