Paul D. Mitchener
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.
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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 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…
The Unitary Conjugation Groupoid as a Universal Mediator of the Baum--Connes Assembly Map
本文展示单位ary共轭群oid作为 Baum--Connes 装配映射的通用中介，扩展了 K_1 指数框架到非 Type I 情况，并通过 Morita 等价将单位ary 共轭群oid与变换群oid联系起来。
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…
A going-down principle for {é}tale groupoids and the Baum-Connes conjecture
本文扩展了étale群胚的下行原理，通过群胚单纯复形研究证明了一般étale群胚的Baum-Connes装配映射分裂单射，并应用于拓扑K-理论连续性和Künneth公式。
A groupoid approach to the equivariant coarse Baum--Connes conjecture
本文通过群体群论方法研究等价粗 Baum--Connes 猜想，引入等价粗群体群并证明其与等价粗 Baum--Connes 猜想的等价性，进一步证明了当空间 X 可以粗嵌入到 Hilbert 空间时，等价粗 Novikov 猜想成立。
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 consider singular foliations whose holonomy groupoid may be nicely decomposed using Lie groupoids (of unequal dimension). We show that the Baum-Connes conjecture can be formulated in this setting. This conjecture is shown to hold under assumptions of amenability. We examine several examples that can be described in this way and make explicit computations of their K-theory.
Categorical approach to the Baum-Connes conjecture for étale groupoids
本文利用等变Kasparov范畴的三角化结构研究étale groupoid的Baum-Connes猜想，证明弱可缩对象与projective对象子范畴互补，提出强Baum-Connes猜想并给出多个推论及应用。
Alternative titles of this paper would have been `Index theory without index' or `The Baum-Connes conjecture without Baum.' In 1989, Rieffel introduced an analytic version of deformation quantization based on the use of continuous fields of C*-algebras. We review how a wide variety of examples of such quantizations can be understood on the basis of a single lemma involving amenable groupoids. These include Weyl-Moyal quantization on manifolds, C*-algebras of Lie groups and Lie groupoids, and the E-theoretic version of the Baum-Connes conjecture for smooth groupoids as described by Connes in hi…
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 an ample groupoid, we construct a spectral sequence with groupoid homology with integer coefficients on the second sheet, converging to the K-groups of the (reduced) groupoid C*-algebra, provided the groupoid has torsion-free stabilizers and satisfies a strong form of the Baum-Connes conjecture. The construction is based on the triangulated category approach to the Baum-Connes conjecture developed by Meyer and Nest. We also present a few applications to topological dynamics and discuss the HK conjecture of Matui.
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…
Chern character for torsion-free ample groupoids
本文构建了从 Baum-Connes 组合映射域到 Crainic-Moerdijk 群oids 带有有理系数的同调群的 Chern 符号映射，验证了 Matui 的 HK 猜想变体，并应用于超几何动力系统、拓扑满群同调等。
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.
We give a new proof of some cases of the Baum-Connes conjecture along the lines of a proof of the Farrell-Jones conjecture.
The Isomorphism Conjectures are translated into the language of homotopical algebra, where they resemble Thomason's descent theorems.
For an $r$-discrete Hausdorff groupoid ${\cal G}$ and an inverse semigroup $S$ of slices of ${\cal G}$ there is an isomorphism between ${\cal G}$-equivariant $KK$-theory and compatible $S$-equivariant $KK$-theory. We use it to define descent homomorphisms for $S$, and indicate a Baum--Connes map for inverse semigroups. Also findings by Khoshkam and Skandalis for crossed products by inverse semigroups are reflected in $KK$-theory.
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…
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.
A Baum-Connes assembly map for essential semigroup crossed products
构造了可分离C*-代数上逆半群Fell丛的等变E-理论和Baum-Connes装配映射，推广了先前工作，并建立了相关函子性质。
The rational HK-conjecture: transformation groupoids and a revised version
本文证明了无挠稳定化子情形下有理HK-猜想对一大类变换群胚成立，并针对有挠情形提出修正版本，利用Baum-Connes猜想与Raven定义的Chern特征作为关键工具证明其适用于Scarparo反例。
We introduce the dual Roe algebras for proper étale groupoid actions and deduce the expected Higson-Roe short exact sequence. When the action is cocompact, we show that the Roe $C^*$-ideal of locally compact operators is Morita equivalent to the reduced $C^*$-algebra of our groupoid, and we further identify the boundary map of the associated periodic six-term exact sequence with the Baum-Connes map, via a Paschke-Higson map for groupoids. For proper actions on continuous families of manifolds of bounded geometry, we associate with any $G$-equivariant Dirac-type family, a coarse index class whi…
Bourbaki seminar 1062, October 2012.
Let F be a global field, A its ring of adeles, G a reductive group over F. We prove the Baum-Connes conjecture for the adelic group G(A).
Coarse geometry of Hecke pairs and the Baum-Connes conjecture
本研究借助Hecke对陪集空间与Schlichting完备化的粗几何，证明了余Haagerup性质下Baum-Connes和Novikov猜想的新稳定性结果，推广了既有结论并给出满足带系数Baum-Connes猜想的S-算术群新实例。
On the Baum-Connes conjecture for $D_{\infty}$
本研究遵循Higson与Kasparov的思路，对无限二面体群的Baum-Connes猜想的证明进行了阐述。
The Baum-Connes conjecture for extensions
本文通过反例证明群扩展中巴乌斯-康纳斯猜想的永久性假设无法简化。
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 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.
We give a survey of the meaning, status and applications of the Baum-Connes Conjecture about the topological K-theory of the reduced group C^*-algebra and the Farrell-Jones Conjecture about the algebraic K- and L-theory of the group ring of a (discrete) group G.
In this note, we exhibit a method to prove the Baum-Connes conjecture (with coefficients) for extensions with finite quotients of certain groups which already satisfy the Baum-Connes conjecture. Interesting examples to which this method applies are torsion-free finite extensions of the pure braid groups, e.g. the full braid groups, or certain fundamental groups of complements of links in S^3.
Let $G$ denote a countable inverse semigroup. We construct a kind of a Baum--Connes map $K(\tilde A \rtimes G) \rightarrow K(A \rtimes G)$ by a categorial approach via localization of triangulated categories, developed by R. Meyer and R. Nest for groups $G$. We allow the coefficient algebras $A$ to be in a special class of algebras called fibered $G$-algebras. This note continues and fixes our preprint "Attempts to define a Baum--Connes map via localization of categories for inverse semigroups".
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 describe a geometric counterpart of the Baum-Connes map for the p-adic group GL(n).
These notes are based on lectures on index theory, topology, and operator algebras at the "School on High Dimensional Manifold Theory" at the ICTP in Trieste, and at the Seminari di Geometria 2002 in Bologna. We describe how techniques coming from the theory of operator algebras, in particular $C^*$-algebras, can be used to study manifolds. Operator algebras are extensively studied in their own right. We will focus on the basic definitions and properties, and on their relevance to the geometry and topology of manifolds. The link between topology and analysis is provided by index theorems. Star…
The main purpose of this paper is to modify the orbit method for the Baum-Connes conjecture as developed by Chabert, Echterhoff and Nest in their proof of the Connes-Kasparov conjecture for almost connected groups \cite{MR2010742} in order to deal with linear algebraic groups over local function fields (i.e., non-archimedean local fields of positive characteristic). As a consequence, we verify the Baum-Connes conjecture for certain Levi-decomposable linear algebraic groups over local function fields. One of these is the Jacobi group, which is the semidirect product of the symplectic group and …
The equivariant coarse Baum-Connes conjecture interpolates between the Baum-Connes conjecture for a discrete group and the coarse Baum-Connes conjecture for a proper metric space. In this paper, we study this conjecture under certain assumptions. More precisely, assume that a countable discrete group $Γ$ acts properly and isometrically on a discrete metric space $X$ with bounded geometry, not necessarily cocompact. We show that if the quotient space $X/Γ$ admits a coarse embedding into Hilbert space and $Γ$ is amenable, and that the $Γ$-orbits in $X$ are uniformly equivariantly coarsely equiva…
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.
On the quantitative coarse Baum-Connes conjecture with coefficients
本文提出了针对真度量空间的带系数定量粗Baum-Connes猜想（QCBC），证明其可由带系数的粗Baum-Connes猜想导出，并指出QCBC最终可归结为一列有界度量空间的一致定量猜想。
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 present a geometric approach to the Baum-Connes conjecture with coefficients for Gromov monster groups via a theorem of Khoskham and Skandalis. Secondly, we use recent results concerning the a-T-menability at infinity of large girth expanders to exhibit a family of coefficients for a Gromov monster group for which the Baum-Connes conjecture is an isomorphism.
We study a group which is hyperbolic relative to a finite family of infinite subgroups. We show that the group satisfies the coarse Baum-Connes conjecture if each subgroup belonging to the family satisfies the coarse Baum-Connes conjecture and admits a finite universal space for proper actions. Especially, the group satisfies the analytic Novikov conjecture.
Every hyperbolic group acts continuously on its Gromov boundary. One can form the corresponding cross-product C*-algebra A. We show that there always exists a canonical Poincare duality map from the K-theory of A to the K-homology of A. We show that this map is an isomorphism when the group in question is the free group on two generators. There is a direct connection between our constructions and the Baum-Connes Conjecture, and we use the latter to deduce our result.
We prove the Baum--Connes conjecture with arbitrary coefficients for some classes of groups: (1) Linear algebraic groups over a non-archimedean local field. (2) Linear algebraic groups over the adeles of a global field k, provided that at every archimedean place of k the associated Lie group is amenable. (3) All closed subgroups of the above groups. This includes linear algebraic groups over global fields - with the same condition as in (2). Update: proof of main results incomplete, maybe not repairable.
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].
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.
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…
Coronae of product spaces and the Coarse Baum-Connes conjecture
本文研究乘积空间与乘积群的coarse Baum-Connes猜想，通过构造并分析proper度量空间乘积的紧化及其边界corona，证明了满足特定假设的CAT(0)群、多项循环群与相对双曲群的乘积满足该猜想。
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 present a history of the Baum-Connes conjecture, the methods involved, the current status, and the mathematics it generated.
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.
We establish a coarse version of the Cartan-Hadamard theorem, which states that proper coarsely convex spaces are coarsely homotopy equivalent to the open cones of their ideal boundaries. As an application, we show that such spaces satisfy the coarse Baum-Connes conjecture. Combined with the result of Osajda-Przytycki, it implies that systolic groups and locally finite systolic complexes satisfy the coarse Baum-Connes conjecture.
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.
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.
Meyer and Nest showed that the Baum--Connes map is equivalent to a map on $K$-theory of two different crossed products. This approach is strongly categorial in method since its bases is to regard Kasparov's theory $KK^G$ as a triangulated category. We have tried to translate this approach to the realm of inverse semigroup equivariant $C^*$-algebras but can prove the existence of a Baum--Connes map only under some unverified additional assumptions which we however strongly motivate. Some of our results may be of independent interest, for example Bott periodicity, the definition of induction fun…
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.
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.
The coarse Baum-Connes conjecture with filtered coefficients and product metric spaces
本文受定量K-理论启发，提出带滤过系数的粗Baum-Connes猜想，推广原始猜想，并证明其满足类在积下封闭，进而发现新的积度量空间例子。
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.
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