The Connes-Kasparov conjecture for almost connected groups

Real versus complex K-theory using Kasparov's bivariant KK-theory

In this paper, we use the KK-theory of Kasparov to prove exactness of sequences relating the K-theory of a real C^*-algebra and of its complexification (generalizing results of Boersema). We use this to relate the real version of the Baum-Connes conjecture for a discrete group to its complex counterpart. In particular, the complex Baum-Connes assembly map is an isomorphism if and only if the real one is, thus reproving a result of Baum and Karoubi. After inverting 2, the same is true for the injectivity or surjectivity part alone.

On the topology of the Kasparov groups and its applications

In this paper we establish a direct connection between stable approximate unitary equivalence for $*$-homomorphisms and the topology of the KK-groups which avoids entirely C*-algebra extension theory and does not require nuclearity assumptions. To this purpose we show that a topology on the Kasparov groups can be defined in terms of approximate unitary equivalence for Cuntz pairs and that this topology coincides with both Pimsner's topology and the Brown-Salinas topology. We study the generalized Rørdam group $KL(A,B)=KK(A,B)/\bar{0}$, and prove that if a separable exact residually finite dime…

Comments on: "Operator $K$-theory for the group SU(n,1)" by P. Julg and G. Kasparov

In this note we point out and fill a gap in the proof by Julg-Kasparov of the Baum-Connes conjecture with coefficients for discrete subgroups of $\op{SU}(n,1)$. The issue at stake is the proof that the complex powers of the contact Laplacian are element of the Heisenberg calculus. In particular, we explain why we cannot implement into the setting of the Heisenberg calculus the classical Seeley's approach to complex powers.

La conjecture de Baum-Connes à coefficients pour les groupes hyperboliques

This paper gives a proof of the Baum-Connes conjecture with coefficients for hyperbolic groups. More precisely the injectivity of the Baum-Connes map was established by Kasparov and Skandalis and we prove the surjectivity.

带系数的巴乌-康纳斯装配映射和米斯钦科-卡斯帕罗夫装配映射对于群扩张

The Baum-Connes and the Mishchenko-Kasparov assembly maps for group extensions

本文研究了带系数的巴乌-康纳斯装配映射e*和米斯钦科-卡斯帕罗夫装配映射μ*在群扩张中的单射性、满射性和同构性，证明了在特定条件下这些映射的性质可以传递到扩展群。

On the Lifting of the Dirac Elements in the Higson-Kasparov Theorem

In this thesis, we investigate the proof of the Baum-Connes Conjecture with Coefficients for a-$T$-menable groups. We will mostly and essentially follow the argument employed by N. Higson and G. Kasparov in the paper [Nigel Higson and Gennadi Kasparov. $E$-theory and $KK$-theory for groups which act properly and isometrically on Hilbert space. Invent. Math., 144(1):23-74, 2001]. The crucial feature is as follows. One of the most important point of their proof is how to get the Dirac elements (the inverse of the Bott elements) in Equivariant $KK$-Theory. We prove that the group homomorphism use…

Universal cycles and homological invariants of locally convex algebras

Using an appropriate notion of locally convex Kasparov modules, we show how to induce isomorphisms under a large class of functors on the category of locally convex algebras; examples are obtained from spectral triples. Our considerations are based on the action of algebraic K-theory on these functors, and involve compatibility properties of the induction process with this action, and with Kasparov-type products. This is based on an appropriate interpretation of the Connes-Skandalis connection formalism. As an application, we prove Bott periodicity and a Thom isomorphism for algebras of Schwar…

Gauge Theory for Spectral Triples and the Unbounded Kasparov Product

We explore factorizations of noncommutative Riemannian spin geometries over commutative base manifolds in unbounded KK-theory. After setting up the general formalism of unbounded KK-theory and improving upon the construction of internal products, we arrive at a natural bundle-theoretic formulation of gauge theories arising from spectral triples. We find that the unitary group of a given noncommutative spectral triple arises as the group of endomorphisms of a certain Hilbert bundle; the inner fluctuations split in terms of connections on, and endomorphisms of, this Hilbert bundle. Moreover, we …

Generalized Connes-Chern characters in KK-theory with an application to weak invariants of topological insulators

We use constructive bounded Kasparov K-theory to investigate the numerical invariants stemming from the internal Kasparov products $K_i(\mathcal A) \times KK^i(\mathcal A, \mathcal B) \rightarrow K_0(\mathcal B) \rightarrow \mathbb R$, $i=0,1$, where the last morphism is provided by a tracial state. For the class of properly defined finitely-summable Kasparov $(\mathcal A,\mathcal B)$-cycles, the invariants are given by the pairing of K-theory of $\mathcal B$ with an element of the periodic cyclic cohomology of $\mathcal B$, which we call the generalized Connes-Chern character. When $\mathcal …

An equivariant orbifold index for proper actions

For a proper, cocompact action by a locally compact group of the form $H \times G$, with $H$ compact, we define an $H \times G$-equivariant index of $H$-transversally elliptic operators, which takes values in $KK_*(C^*H, C^*G)$. This simultaneously generalises the Baum--Connes analytic assembly map, Atiyah's index of transversally elliptic operators, and Kawasaki's orbifold index. This index also generalises the assembly map to elliptic operators on orbifolds. In the special case where the manifold in question is a real semisimple Lie group, $G$ is a cocompact lattice and $H$ is a maximal comp…

On the Baum--Connes conjecture for discrete quantum groups with torsion and the quantum Rosenberg Conjecture

We give a decomposition of the equivariant Kasparov category for discrete quantum group with torsions. As an outcome, we show that the crossed product by a discrete quantum group in a certain class preserves the UCT. We then show that quasidiagonality of a reduced C*-algebra of a countable discrete quantum group $Γ$ implies that $Γ$ is amenable, and deduce from the work of Tikuisis, White and Winter, and the results in the first part of the paper, the converse (i.e. the quantum Rosenberg Conjecture) for a large class of countable discrete unimodular quantum groups. We also note that the unimod…

On the Baum-Connes Conjecture for Groups Acting on CAT(0)-Cubical Spaces

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.

A transverse Chern character for compact Lie group actions

This article aims to explore new perspectives offered by Kasparov's recent work on transverse index theory in the context of actions of compact Lie groups on a manifold, and hints at potential connections to the work of Berline-Vergne and Paradan-Vergne.

张量范畴等变 KK-理论

Tensor category equivariant KK-theory

本文构建 C*-张量范畴对称性下的等变 Kasparov 双变量 K-理论，证明其保留核心结构并具有弱 Morita 等价不变性，还考察了离散群 3-上循环扭曲的 Baum-Connes 型性质。

Weyl character formula in KK-theory

The purpose of this paper is to begin an exploration of connections between the Baum-Connes conjecture in operator $\K$-theory and the geometric representation theory of reductive Lie groups. Our initial goal is very modest, and we shall not stray far from the realm of compact groups, where geometric representation theory amounts to elaborations of the Weyl character formula such as the Borel-Weil-Bott theorem. We shall recast the topological $\K$-theory approach to the Weyl character formula, due basically to Atiyah and Bott, in the language of Kasparov's $\K\K$-theory. Then we shall show how…

A Going-Down principle for ample groupoids and 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…

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 equival…

On an intermediate bivariant theory for $C^*$-algebras, I

We construct a new bivariant theory, that we call $KE$-theory, which is intermediate between the $KK$-theory of G. G. Kasparov, and the $E$-theory of A. Connes and N. Higson. For each pair of separable graded $C^*$-algebras $A$ and $B$, acted upon by a locally compact $σ$-compact group $G$, we define an abelian group $KE_G(A,B)$. We show that there is an associative product $KE_G(A,D) \otimes KE_G(D,B) \to KE_G(A,B)$. Various functoriality properties of the $KE$-theory groups and of the product are presented. The new theory has a simpler product than $KK$-theory and there are natural transform…

Compact operators and algebraic $K$-theory for groups which act properly and isometrically on Hilbert space

We prove the $K$-theoretic Farrell-Jones conjecture for groups as in the title with coefficient rings and $C^*$-algebras which are stable with respect to compact operators. We use this and Higson-Kasparov's result that the Baum-Connes conjecture with coefficients holds for such groups, to show that if $G$ is as in the title then the algebraic and the $C^*$-crossed products of $G$ with a stable $C^*$-algebra have the same $K$-theory.

The Novikov conjecture, the group of volume preserving diffeomorphisms and Hilbert-Hadamard spaces

We prove that the Novikov conjecture holds for any discrete group admitting an isometric and metrically proper action on an admissible Hilbert-Hadamard space. Admissible Hilbert-Hadamard spaces are a class of (possibly infinite-dimensional) non-positively curved metric spaces that contain dense sequences of closed convex subsets isometric to Riemannian manifolds. Examples of admissible Hilbert-Hadamard spaces include Hilbert spaces, certain simply connected and non-positively curved Riemannian-Hilbertian manifolds and infinite\-/dimensional symmetric spaces. Thus our main theorem can be consid…

étale groupoid的Baum-Connes猜想的范畴方法

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。

The C(X)-algebra of a net and index theory

Given a connected and locally compact Hausdorff space X with a good base K we assign, in a functorial way, a C(X)-algebra to any precosheaf of C*-algebras A defined over K. Afterwards we consider the representation theory and the Kasparov K-homology of A, and interpret them in terms, respectively, of the representation theory and the K-homology of the associated C(X)-algebra. When A is an observable net over the spacetime X in the sense of algebraic quantum field theory, this yields a geometric description of the recently discovered representations affected by the topology of X.

KK-theoretic duality for proper twisted actions

Let the discrete group G act properly and isometrically on the Riemannian manifold X. Let C_0(X, δ) be the section algebra of a smooth locally trivial G-equivariant bundle of elementary C*-algebras representing an element δof the Brauer group Br_G(X). Then C_0(X,δ^{-1}) x G is KK-theoretically Poincare dual to (C_0(X,δ)\otimes_{C_0(X)} C_τ(X)) xG, where δ^{-1} is the inverse of δin the Brauer group. We deduce this from a strengthening of Kasparov's duality theorem RKK^G(X; A,B) \cong KK^G(C_τ(X)\otimes A, B). As applications we also obtain a version of the above Poincare duality with X replace…

On the spectral sequence associated with the Baum-Connes Conjecture for $\mathbb Z^n$

We examine a spectral sequence that is naturally associated with the Baum-Connes Conjecture with coefficients for $\mathbb Z^n$ and also constitutes an instance of Kasparov's construction in his work on equivariant $KK$-theory. For $k\leq n$, we give a partial description of the $k$-th page differential of this spectral sequence, which takes into account the natural $\mathbb Z^k$-subactions. In the special case that the action is trivial in $K$-theory, the associated second page differential is given by a formula involving the second page differentials of the canonical $\mathbb Z^2$-subactions…

几乎连通李群的$K$-同调欧拉类的等变局域化

Equivariant Localization of $K$-homological Euler Class for almost connected Lie Groups

该研究针对几乎连通李群，利用Witten形变与局域化代数技术计算等变$K$-同调类，通过Morse-Bott扰动实现类的局域化，得到等变庞加莱-霍普夫公式并提供等变指标计算工具。