Given an action of a discrete quantum group (in the sense of Van Daele, Kustermans and Effros-Ruan)
${\cal A}$ on a $C^*$-algebra ${\cal C}$, satisfying some regularity assumptions resembling the proper $Γ$-compact action for a classical discrete group $Γ$ on some space, we are able to construct canonical maps $μ^i_r$ (\$μ_i$ respectively) ($i=0,1$) from the ${\cal A}$-equivariant K-homology groups $KK_i^{\cal A}({\cal C}, C |)$ to the K-theory groups $K_i(\hat{{\cal A}_r})$ ($K_i(\hat{\cal A})$ respectively), where $\hat{{\cal A}_r}$ and $\hat{\cal A}$ stand for the quantum analogues of the reduced and full group $C^*$-algebras (c.f. [11],[6]). We follow the steps of the construction of the classical Baum-Connes map, although in the context of quantum group the nontrivial modular property of the invariant weights (and the related fact that the square of the antipode is not identity) has to be taken into serious consideration, making it somewhat tricky to guess and prove the correct definitions of relevant Hilbert module structures.
We define and study an analogue of the Baum-Connes assembly map for complex semisimple quantum groups, that is, Drinfeld doubles of $ q $-deformations of compact semisimple Lie groups. Our starting point is the deformation picture of the Baum-Connes assembly map for a complex semisimple Lie group $ G $, which allows one to express the $ K $-theory of the reduced group $ C^* $-algebra of $ G $ in terms of the $ K $-theory of its associated Cartan motion group. The latter can be identified with the semidirect product of the maximal compact subgroup $ K $ acting on $ \mathfrak{k}^* $ via the coad…
The Baum-Connes conjecture predicts that a certain assembly map is an isomorphism. We identify the homotopy theoretical construction of the assembly map by Davis and Lück with the category theoretical construction by Meyer and Nest. This extends the result of Hambleton and Pedersen to arbitrary coefficients. Our approach uses abstract properties rather than explicit constructions and is formally similar to Meyer's and Nest's identification of their assembly map with the original construction of the assembly map by Baum, Connes and Higson.
We show that the real Baum-Connes conjecture for abelian groups, possibly twisted by a cocycle, explains the isomorphisms of (twisted) KR-groups that underlie all T-dualities of torus orientifold string theories.
We show that the image of Connes-Karoubi-Chern character, restricted to the image of the Baum-Connes assembly map in the Bott-periodized topological K-theory of the complex group algebra, lies in the elliptic summand of the (periodic) cyclic homology of the group algebra. This implies that for any (weighted) ell-1 completion of the group algebra, rational surjectivity of the Baum-Connes assembly map implies the generalized Bass conjecture for that algebra.
In this work, the continuously controlled techniques developed by Carlsson and Pedersen are used to prove that the Baum-Connes map is a split injection for groups satisfying certain geometric conditions.
In this note we set a configuration space description of the equivariant connective K-homology groups with coefficients in a unital C*-algebra for proper actions. Over this model we define a connective assembly map and prove that in this setting is possible to recover the analytic assembly map.
Using a combination of Atiyah-Segal ideas on one side and of Connes and Baum-Connes ideas on the other, we prove that the Twisted geometric K-homology groups of a Lie groupoid have an external multiplicative structure extending hence the external product structures for proper cases considered by Adem-Ruan in [1] or by Tu,Xu and Laurent-Gengoux in [24]. These Twisted geometric K-homology groups are the left hand sides of the twisted geometric Baum-Connes assembly maps recently constructed in [9] and hence one can transfer the multiplicative structure via the Baum-Connes map to the Twisted K-the…
We study in this paper the maximal version of the coarse Baum-Connes assembly map for families of expanding graphs arising from residually finite groups. Unlike for the usual Roe algebra, we show that this assembly map is closely related to the (maximal) Baum-Connes assembly map for the group and is an isomorphism for a class of expanders. We also introduce a quantitative Baum-Connes assembly map and discuss its connections to K-theory of (maximal) Roe algebras.
We redefine the Baum-Connes assembly map using simplicial approximation in the equivariant Kasparov category. This new interpretation is ideal for studying functorial properties and gives analogues of the assembly maps for all equivariant homology theories, not just for the K-theory of the crossed product. We extend many of the known techniques for proving the Baum-Connes conjecture to this more general setting.
We construct the geometric Baum-Connes assembly map for twisted Lie groupoids, that means for Lie groupoids together with a given groupoid equivariant $PU(H)-$principle bundle. The construction is based on the use of geometric deformation groupoids, these objects allow in particular to give a geometric construction of the associated pushforward maps and to establish the functoriality. The main results in this paper are to define the geometric twisted K-homology groups and to construct the assembly map. Even in the untwisted case the fact that the geometric twisted K-homology groups and the geo…
Given a (not necessarily discrete) proper metric space $M$ with bounded geometry, we define a groupoid $G(M)$. We show that the coarse Baum--Connes conjecture with coefficients, which states that the assembly map with coefficients for G(M) is an isomorphism, is hereditary by taking closed subspaces.
We document some versions, in real K-theory, of well-known properties of the coarse assembly map in complex K-theory. These results are well-known, but difficult to find in the literature.
We introduce the notion of proper Kasparov cycles for Kasparov's G-equivariant KK-theory for a general locally compact, second countable topological group G. We show that for any proper Kasparov cycle, its induced map on K-theory factors through the left-hand side of the Baum-Connes conjecture. This allows us to upgrade the direct splitting method, a recent new approach to the Baum-Connes conjecture which, in contrast to the standard gamma element method (the Dirac dual-Dirac method), avoids the need of constructing proper algebras and the Dirac and the dual-Dirac elements. We introduce the no…
For a countable discrete group $G$, we construct a new and concrete model for the equivariant topological $K$-homology theory of $G$, which is defined for all $G$-actions, not just for proper $G$-actions. The construction of our model combines some of the ideas from the localization algebra approach to the coarse Baum-Connes conjecture and the controlled algebra approach to the Farrell-Jones conjecture. It brings new ideas into the study of the Baum-Connes conjecture. First, as the model is defined for all $G$-actions, we are able to define relative assembly maps. We prove, by using an inducti…
The Baum-Connes map for finitely generated free abelian groups is a K-theoretic analogue of the Fourier-Mukai transform from algebraic geometry. We describe this K-theoretic transform in the language of topological correspondences, and compute its action on K-theory (of tori) described geometrically in terms of Baum-Douglas cocycles, showing that the Fourier-Mukai transform maps the class of a subtorus to the class of a suitably defined dual torus. We deduce the Fourier-Mukai inversion formula. We use these results to give a purely geometric description of the Baum-Connes assembly map for free…
In this article we give a characterisation of the Baum-Connes assembly map with coefficients. The technical tools needed are the K-theory of C*-categories, and equivariant KK-theory in the world of groupoids.
This paper studies higher index theory for a random sequence of bounded degree, finite graphs with diameter tending to infinity. We show that in a natural model for such random sequences the following hold almost surely: the coarse Baum-Connes assembly map is injective; the coarse Baum-Connes assembly map is not surjective; the maximal coarse Baum-Connes assembly map is an isomorphism. These results are closely tied to issues of expansion in graphs: in particular, we also show that such random sequences almost surely do not have geometric property (T), a strong form of expansion. The key geome…
We construct a Baum--Connes assembly map localised at the unit element of a discrete group $Γ$. This morphism, called $μ_τ$, is defined in $KK$-theory with coefficients in $\mathbb{R}$ by means of the action of the projection $[τ]\in KK_{\mathbb{R}}^Γ(\mathbb{C},\mathbb{C})$ canonically associated to the group trace of $Γ$. We show that the corresponding $τ$-Baum--Connes conjecture is weaker then the classical one but still implies the strong Novikov conjecture. The right hand side of $μ_τ$ is functorial with respect to the group $Γ$.
The well known "associativity property" of the crossed product by a semi-direct product of discrete groups is generalized into the context of discrete \emph{quantum} groups. This decomposition allows to define an appropriate triangulated functor relating the Baum-Connes property for the quantum semi-direct product to the Baum-Connes property for the discrete quantum groups involved in the construction. The corresponding stability result for the Baum-Connes property generalizes a result of J. Chabert for a quantum semi-direct product under torsion-freeness assumption. The $K$-amenability connex…
We develop a generalization of quantitative $K$-theory, which we call controlled $K$-theory. It is powerful enough to study the $K$-theory of crossed product of $C^*$-algebras by action of étale groupoids and discrete quantum groups. In this article, we will use it to study groupoids crossed products. We define controlled assembly maps, which factorize the Baum-Connes assembly maps, and define the controlled Baum-Connes conjecture. We relate the controlled conjecture for groupoids to the classical conjecture, and to the coarse Baum-Connes conjecture. This allows to give applications to Coarse …
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 $\ell_2$-Betti numbers of a group and surjectivity of different Baum-Connes type assembly maps.
In this paper, the first of a series of two, we continue the study of higher index theory for expanders. We prove that if a sequence of graphs is an expander and the girth of the graphs tends to infinity, then the coarse Baum-Connes assembly map is injective, but not surjective, for the associated metric space $X$. Expanders with this girth property are a necessary ingredient in the construction of the so-called `Gromov monster' groups that (coarsely) contain expanders in their Cayley graphs. We use this connection to show that the Baum-Connes assembly map with certain coefficients is injectiv…
We define and compare two bivariant generalizations of the topological $K$-group $K^\top(G)$ for a topological group $G$. We consider the Baum-Connes conjecture in this context and study its relation to the usual Baum-Connes conjecture.
We show that complex semisimple quantum groups, that is, Drinfeld doubles of $ q $-deformations of compact semisimple Lie groups, satisfy a categorical version of the Baum-Connes conjecture with trivial coefficients. This approach, based on homological algebra in triangulated categories, is compatible with the previously studied deformation picture of the assembly map, and allows us to define an assembly map with arbitrary coefficients for these quantum groups.
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.
We show that the Baum-Connes morphism twisted by a non-unitary representation, defined in [GA08], is an isomorphism for a large class of groups satisfying the Baum-Connes conjecture. Such class contains all the real semi-simple Lie groups, all hyperbolic groups and many infinite discret groups having Kazhdan's property (T). We define a tensorisation by a non-unitary finite dimensional representation on the left handside of the Baum-Connes morphism and we show that its analogue in K-theory must be defined on the K-theory of the twisted group algebras introduced in [GA07b].
In this article we prove that the $KH$-asembly map, as defined by Bartels and L{ü}ck, can be described in terms of the algebraic $KK$-theory of Cortinas and Thom. The $KK$-theory description of the $KH$-assembly map is similar to that of the Baum-Connes assembly map. In very elementary cases, methods used to prove the Baum-Connes conjecture also apply to the $KH$-isomorphism conjecture.
We review the formulation and proof of the Baum-Connes conjecture for the dual of the quantum group $ SU_q(2) $ of Woronowicz. As an illustration of this result we determine the $ K $-groups of quantum automorphism groups of simple matrix algebras.
We introduce a new method for studying the Baum-Connes conjecture, which we call the direct splitting method. The method can simplify and clarify proofs of some of the known cases of the conjecture. In a separate paper, with J. Brodzki, E. Guentner and N. Higson, a similar idea will be used to give a finite-dimensional proof of the Baum-Connes conjecture for groups which act properly and co-compactly on a finite-dimensional CAT(0)-cubical space.
We give a complete solution, for discrete countable groups, to the problem of defining and computing a geometric pairing between the left hand side of the Baum-Connes assembly map, given in terms of geometric cycles associated to proper actions on manifolds, and cyclic periodic cohomology of the group algebra. Indeed, for any such group $Γ$ (without any further assumptions on it) we construct an explicit morphism from the Left-Hand side of the Baum-Connes assembly map to the periodic cyclic homology of the group algebra. This morphism, called here the Chern-Baum-Connes assembly map, allows to …
We prove that the coarse assembly maps for proper metric spaces which are non-positively curved in the sense of Busemann are isomorphisms, where we do not assume that the spaces are with bounded coarse geometry. Also it is shown that we can calculate the coarse K-homology and the K-theory of the Roe algebra by using the visual boundaries.
In this paper, we verify the $L^p$ coarse Baum-Connes conjecture for spaces with finite asymptotic dimension for $p\in[1,\infty)$. We also show that the $K$-theory of $L^p$ Roe algebras are independent of $p\in(1,\infty)$ for spaces with finite asymptotic dimension.
Let G be a locally compact group and rho a non-unitary finite dimensional representation of G. We consider tensor products of rho by some unitary representations of G in order to define two Banach algebras analogous to the group C*-algebras, C*(G) and C*_r(G). We calculate the K-theory of such algebras for a large class of groups satisfying the Baum-Connes conjecture.
We study a Going-Down (or restriction) principle for ample groupoids and its applications. The Going-Down principle for locally compact groups was developed by Chabert, Echterhoff and Oyono-Oyono and allows to study certain functors, that arise in the context of the topological K-theory of a locally compact group, in terms of their restrictions to compact subgroups. We extend this principle to the class of ample Hausdorff groupoids using Le Gall's groupoid equivariant version of Kasparov's bivariant KK-theory. Moreover, we provide an application to the Baum-Connes conjecture for ample groupoid…
Using the unbounded picture of analytical K-homology, we associate a well-defined K-homology class to an unbounded symmetric operator satisfying certain mild technical conditions. We also establish an ``addition formula'' for the Dirac operator on the circle and for the Dolbeault operator on closed surfaces. Two proofs are provided, one using topology and the other one, surprisingly involved, sticking to analysis, on the basis of the previous result. As a second application, we construct, in a purely analytical language, various homomorphisms linking the homology of a group in low degree, the …
We introduce a new variant of the coarse Baum-Connes conjecture designed to tackle coarsely disconnected metric spaces called the boundary coarse Baum-Connes conjecture. We prove this conjecture for many coarsely disconnected spaces that are known to be counterexamples to the coarse Baum-Connes conjecture. In particular, we give a geometric proof of this conjecture for spaces of graphs that have large girth and bounded vertex degree. We then connect the boundary conjecture to the coarse Baum-Connes conjecture using homological methods, which allows us to exhibit all the current uniformly discr…
In this paper we consider the Baum-Connes correspondence for the affine and extended affine Weyl groups of a compact connected semisimple Lie group. We show that the Baum-Connes correspondence in this context arises from Langlands duality for the Lie group.
The goal of this article is to provide a bridge between the gamma element method for the Baum--Connes conjecture (the Dirac dual-Dirac method) and the controlled algebraic approach of Roe and Yu (localization algebras). For any second countable, locally compact group G, we study the reduced crossed product algebras of the representable localization algebras for proper G-spaces. We show that the naturally defined forget-control map is equivalent to the Baum--Connes assembly map for any locally compact group G and for any coefficient G-C*-algebra B. We describe the gamma element method for the B…
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 equival…
We prove that if the classical Baum-Connes conjecture in complex K-theory is true (for a given discrete group G), then the conjecture is also true in the real case (for the same group G). The essential ingredients of the proof are the descent theorem in topological K-theory (math.KT/0509396) and a Paschke duality for C*-algebras proved by John Roe (Q.J. Math. 55 (2004) 325-331). According to S. Stolz, this result is linked with the existence of a Riemannian metric of positive scalar curvature on a compact connected spin manifold with G as fundamental group.
We reformulate the Baum-Connes conjecture with coefficients by introducing a new crossed product functor for C*-algebras. All confirming examples for the original Baum-Connes conjecture remain confirming examples for the reformulated conjecture, and at present there are no known counterexamples to the reformulated conjecture. Moreover, some of the known expander-based counterexamples to the original Baum-Connes conjecture become confirming examples for our reformulated conjecture.
We introduce and analyze the concept of an assembly map from the original homotopy theoretic point of view. We give also interpretations in terms of surgery theory, controlled topology and index theory. The motivation is that prominent conjectures of Farrell-Jones and Baum-Connes about K- and L-theory of group rings and group C^*-algebras predict that certain assembly maps are weak homotopy equivalences.
We give a new proof of the Baum--Connes conjecture with coefficients for any second countable, locally compact topological group that acts properly and cocompactly on a finite-dimensional CAT(0)-cubical space with bounded geometry. The proof uses the Julg-Valette complex of a CAT(0)-cubical space introduced by the first three authors, and the direct splitting method in Kasparov theory developed by the last author.
In this paper we study the index theoretic interpretation of the analytical assembly map that appears in the Baum-Connes conjecture. In its general form it may be constructed using Kasparov's equivariant KK-theory. In the special case of a torsionfree group the domain simplifies to the usual K-homology of the classifying space BG of G and it is frequently used that in this case the analytical assembly map is given by assigning to an operator an equivariant index. We give a precise formulation of this statement and prove it.
