Let $A$ be a separable $C^*$-algebra and $B$ a stable $C^*$-algebra containing a strictly positive element. We show that the group $\Ext(SA,B)$ of unitary equivalence classes of extensions of $SA$ by $B$, modulo the extensions which are asymptotically split, coincides with the group of homotopy classes of such extensions. This is done by proving that the Connes-Higson construction gives rise to an isomorphism between $\Ext(SA,B)$ and the $E$-theory group $E(A,B)$ of homotopy classes of asymptotic homomorphisms from $S^2A$ to $B$.
Alain Connes and Nigel Higson pointed out in the 1990s that the Connes-Kasparov "conjecture"' for the K-theory of reduced groupe $C^\ast$-algebras seemed, in the case of reductive Lie groups, to be a cohomological echo of a conjecture of George Mackey concerning the rigidity of representation theory along the deformation from a reductive Lie group to its Cartan motion group. For complex semisimple groups, Nigel Higson established in 2008 that Mackey's analogy is a real phenomenon and does lead to a simple proof of the Connes-Kasparov isomorphism. We here turn to more general reductive groups a…
We construct an isomorphism between the geometric model and Higson-Roe's analytic surgery group, reconciling the constructions in the previous papers in the series on "Realizing the analytic surgery group of Higson and Roe geometrically" with their analytic counterparts. Following work of Lott and Wahl, we construct a Chern character on the geometric model for the surgery group; it is a "delocalized Chern character", from which Lott's higher delocalized $ρ$-invariants can be retrieved. Following work of Piazza and Schick, we construct a geometric map from Stolz' positive scalar curvature seque…
Let $\widetilde{X}$ be a smooth Riemannian manifold equipped with a proper, free, isometric and cocompact action of a discrete group $Γ$. In this paper we prove that the analytic surgery exact sequence of Higson-Roe for $\widetilde{X}$ is isomorphic to the exact sequence associated to the adiabatic deformation of the Lie groupoid $\widetilde{X}\times_Γ\widetilde{X}$. We then generalize this result to the context of smoothly stratified manifolds. Finally, we show, by means of the aforementioned isomorphism, that the $\varrho$-classes associated to a metric with positive scalar curvature defined…
In the 1970's, George Mackey pointed out an analogy that exists between tempered representations of semisimple Lie groups and unitary representations of associated semidirect product Lie groups. More recently, Nigel Higson refined Mackey's analogy into a one-to-one correspondence for connected complex semisimple groups, and in doing so obtained a novel verification of the Baum-Connes conjecture with trivial coefficients for such groups. Here we extend Higson's results to any Lie group with finitely many connected components whose connected component of the identity is complex semisimple. Our m…
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…
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…
The algebra $Ψ(M)$ of order zero pseudodifferential operators on a compact manifold $M$ defines a well-known $C^*$-extension of the algebra $C(S^*M)$ of continuous functions on the cospherical bundle $S^*M\subset T^*M$ by the algebra $\K$ of compact operators. In his proof of the index theorem, Higson defined and used an asymptotic homomorphism $T$ from $C_0(T^*M)$ to $\K$, which plays the role of a deformation for the commutative algebra $C_0(T^*M)$. Similar constructions exist also for operators and symbols with coefficients in a $C^*$-algebra. We show that the image of the above extension u…
Let $A$, $B$ be C*-algebras; $A$ separable, $B$ $σ$-unital and stable. We introduce a notion of translation invariance for asymptotic homomorphisms from $SA=C_0(\mathbb R)\otimes A$ to $B$ and show that the Connes-Higson construction applied to any extension of $A$ by $B$ is homotopic to a translation invariant asymptotic homomorphism. In the other direction we give a construction which produces extensions of $A$ by $B$ out of such a translation invariant asymptotic homomorphism. This leads to our main result; that the homotopy classes of extensions coincide with the homotopy classes of transl…
Let $A$ be a separable $C^*$-algebra and let $B$ be a stable $C^*$-algebra with a strictly positive element. We consider the (semi)group $\Ext^{as}(A,B)$ (resp. $\Ext(A,B)$) of homotopy classes of asymptotic (resp. of genuine) homomorphisms from $A$ to the corona algebra $M(B)/B$ and the natural map $i:\Ext(A,B)\ar\Ext^{as}(A,B)$. We show that if $A$ is a suspension then $\Ext^{as}(A,B)$ coincides with $E$-theory of Connes and Higson and the map $i$ is surjective. In particular any asymptotic homomorphism from $SA$ to $M(B)/B$ is homotopic to some genuine homomorphism.
Let $G$ be a group and let $E$ be a functor from small $\Z$-linear categories to spectra. Also let $A$ be a ring with a $G$-action. Under mild conditions on $E$ and $A$ one can define an equivariant homology theory of $G$-simplicial sets $H^G(-,E(A))$ with the property that if $H\subset G$ is a subgroup, then \[ H^G_*(G/H,E(A))=E_*(A\rtimes H) \] If now $\cF$ is a nonempty family of subgroups of $G$, closed under conjugation and under subgroups, then there is a model category structure on $G$-simplicial sets such that a map $X\to Y$ is a weak equivalence (resp. a fibration) if and only if $X^H…
We show that the E-theory of Connes and Higson can be formulated in terms of C*-extensions in a way quite similar to the way in which the KK-theory of Kasparov can. The essential difference is that the role played by split extensions should be taken by asymptotically split extensions. We call an extension of a C*-algebra $A$ by a stable C*-algebra $B$ asymptotically split if there exists an asymptotic homomorphism consisting of right inverses for the quotient map. An extension is called semi-invertible if it can be made asymptotically split by adding another extension to it. Our main result is…
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…
Using a smooth version of the Connes--Thom isomorphism in Grensing's bivariant K-theory for locally convex algebras, we prove an equivariant version of the Connes--Thom isomorphism in periodic cyclic homology. As an application, we prove that periodic cyclic homology is invariant with respect to equivariant strict deformation quantization.
Let $A$, $A'$ be separable $C^*$-algebras, $B$ a stable $σ$-unital $C^*$-algebra. Our main result is the construction of the pairing $[[A',A]]\times\operatorname{Ext}^{-1/2}(A,B)\to\operatorname{Ext}^{-1/2}(A',B)$, where $[[A',A]]$ denotes the set of homotopy classes of asymptotic homomorphisms from $A'$ to $A$ and $\operatorname{Ext}^{-1/2}(A,B)$ is the group of semi-invertible extensions of $A$ by $B$. Assume that all extensions of $A$ by $B$ are semi-invertible. Then this pairing allows us to give a condition on $A'$ that provides semi-invertibility of all extensions of $A'$ by $B$. This ho…
Let $G$ be a locally compact group with cocompact connected component. We prove that the assembly map from the topological $\k$-theory of $G$ to the $\k$-theory of the reduced $C^*$-algebra of $G$ is an isomorphism.
Let $A \in M_{n}(\mathbb{R})$ be an invertible matrix. Consider the semi-direct product $\mathbb{R}^{n} \rtimes \mathbb{Z}$ where $\mathbb{Z}$ acts on $\mathbb{R}^{n}$ by matrix multiplication. Consider a strongly continuous action $(α,τ)$ of $\mathbb{R}^{n} \rtimes \mathbb{Z}$ on a $C^{*}$-algebra $B$ where $α$ is a strongly continuous action of $\mathbb{R}^{n}$ and $τ$ is an automorphism. The map $τ$ induces a map $\widetildeτ$ on $B \rtimes_α \mathbb{R}^{n}$. We show that, at the $K$-theory level, $τ$ commutes with the Connes-Thom map if $\det(A)>0$ and anticommutes if $\det(A)<0$. As an ap…
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.
In this paper we construct a Chern-Weil isomorphism for the equivariant Brauer group of R^n-actions on a principal torus bundle, where the target for this isomorphism is a "dimensionally reduced" Cech cohomology group. From this point of view, the usual forgetful functor takes the form of a connecting homomorphism in a long exact sequence in dimensionally reduced cohomology.
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…
For an orbifold X and $α\in H^3(X, Z)$, we introduce the twisted cohomology $H^*_c(X, α)$ and prove that the Connes-Chern character establishes an isomorphism between the twisted K-groups $K_α^* (X) \otimes C$ and twisted cohomology $H^*_c(X, α)$. This theorem, on the one hand, generalizes a classical result of Baum-Connes, Brylinski-Nistor, and others, that if X is an orbifold then the Chern character establishes an isomorphism between the K-groups of X tensored with C, and the compactly-supported cohomology of the inertia orbifold. On the other hand, it also generalizes a recent result of Ad…
