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 dimensional C*-algebra satisfies the universal coefficient theorem in KK-theory, then it embeds in the UHF algebra of type $2^\infty$.
In particular such an embedding exists for the C*-algebra of a second countable amenable locally compact maximally almost periodic group.
In this paper it is demonstrated that the Kasparov pairing is continuous with respect to the natural topology on the Kasparov groups, so that a KK-equivalence is an isomorphism of topological groups. In addition, we demonstrate that the groups have a natural pseudopolonais structure, and we prove that various KK-structural maps are continuous.
Let G be a finite group. We systematically exploit general homological methods in order to reduce the computation of G-equivariant KK-theory to topological equivariant K-theory. The key observation is that the functor assigning to a separable G-C*-algebra the collection of all its equivariant K-theory groups lifts naturally to a homological functor taking values in the abelian tensor category of Mackey modules over the classical representation Green functor for G. This fact yields a new universal coefficient and a new Kuenneth spectral sequence for the G-equivariant Kasparov category, whose co…
The Kasparov groups KK_*(A, B) have a natural structure as pseudopolonais groups. In this paper we analyze how this topology interacts with the terms of the Universal Coefficient Theorem (UCT) and the splittings of the UCT constructed by J. Rosenberg and the author, as well as its canonical three term decomposition which exists under bootstrap hypotheses. We show that the various topologies on Ext_{\Bbb Z}^1(K_*(A), K_*(B)) and other related groups mostly coincide. Then we focus attention on the Milnor sequence and the fine structure subgroup of KK_*(A, B). An important consequence of our work…
We construct a Fredhom module representing the Kasparov gamma element in G-equivariant KK-theory for G a semisimple Lie group of real rank one. This is the main step of our proof of the Baum-Connes conjecture for such groups.
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.
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 G be a locally compact group. We describe elements of KK^G (A,B) by equivariant homomorphisms, following Cuntz's treatment in the non-equivariant case. This yields another proof for the universal property of KK^G: It is the universal split exact stable homotopy functor. To describe a Kasparov triple (E, phi, F) by an equivariant homomorphism, we have to arrange for the Fredholm operator F to be equivariant. This can be done if A is of the form K(L^2G) otimes A' and more generally if the group action on A is proper in the sense of Rieffel and Exel.
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…
The bulk-boundary and a new bulk-defect correspondence principles are formulated using groupoid algebras. The new strategy relies on the observation that the groupoids of lattices with boundaries or defects display spaces of units with invariant accumulation manifolds, hence they can be naturally split into disjoint unions of open and closed invariant sub-sets. This leads to standard exact sequences of groupoid $C^\ast$-algebras that can be used to associate a Kasparov element to a lattice defect and to formulate an extremely general bulk-defect correspondence principle. As an application, we …
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…
We introduce a notion of Morita equivalence for non-selfadjoint operator algebras equipped with a completely isometric involution (operator *-algebras). We then show that the unbounded Kasparov product by a Morita equivalence bimodule induces an isomorphism between equivalence classes of twisted spectral triples over Morita equivalent operator *-algebras. This leads to a tentative definition of unbounded bivariant K-theory and we prove that this bivariant theory is related to Kasparov's bivariant K-theory via the Baaj-Julg bounded transform. Moreover, the unbounded Kasparov product provides a …
For C*-algebras $A$ and $B$, we generalize the notion of a quasihomomorphism from $A$ to $B$, due to Cuntz, by considering quasihomomorphisms from some C*-algebra $C$ to $B$ such that $C$ surjects onto $A$, and the two maps forming a quasihomomorphism agree on the kernel of this surjection. Under an additional assumption, the group of homotopy classes of such generalized quasihomomorphisms coincides with $KK(A,B)$. This makes the definition of Kasparov's bifunctor slightly more symmetric and gives more flexibility for constructing elements of $KK$-groups. These generalized quasihomomorphisms c…
We review the theory of Toeplitz extensions and their role in operator K-theory, including Kasparov's bivariant K-theory. We then discuss the recent applications of Toeplitz algebras in the study of solid state systems, focusing in particular on the bulk-edge correspondence for topological insulators.
In this paper we identify QD(A,B), the quasidiagonal classes in KK_1(A,B), in terms of K_*(A) and K_*(B), and we use these results in various applications. Here is our central result. Theorem: Suppose that A is in the category of separable nuclear C^*-algebras which satisfy the UCT and A is quasidiagonal relative to B. Then there is a natural isomorphism QD(A,B) = Pext (K_*(A), K_*(B))_0 . Thus quasidiagonality of KK-classes is indeed a topological invariant. We give several applications. Finally, we establish a converse to a theorem of Davidson, Herrero, and Salinas, giving conditions under w…
We have been studying the index theory for some special infinite-dimensional manifolds with a "proper cocompact" actions of the loop group LT of the circle T, from the viewpoint of the noncommutative geometry. In this paper, we will introduce the LT-equivariant KK-theory and we will construct three KK-elements: the index element, the Clifford symbol element and the Dirac element. These elements satisfy a certain relation, which should be called the (KK-theoretical) index theorem, or the KK-theoretical Poincaré duality for infinite-dimensional manifolds. We will also discuss the assembly maps.
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 …
Let a compact group G act on real or complex C*-algebras A and B, with A separable and B sigma-unital. We express the G-equivariant Kasparov groups KK_n(A,B) by algebraic K-groups of a certain additive category.
We propose a new notion of unbounded $K\!K$-cycle, mildly generalising unbounded Kasparov modules, for which the direct sum is well-defined. To a pair $(A,B)$ of $σ$-unital $C^{*}$-algebras, we can then associate a semigroup $\overline{U\!K\!K}(A,B)$ of homotopy equivalence classes of unbounded cycles, and we prove that this semigroup is in fact an abelian group. In case $A$ is separable, our group $\overline{U\!K\!K}(A,B)$ is isomorphic to Kasparov's $K\!K$-theory group $K\!K(A,B)$ via the bounded transform. We also discuss various notions of degenerate cycles, and we prove that the homotopy …
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.
We construct equivariant $KK$-theory with coefficients in $\mathbb{R}$ and $\mathbb{R}/\mathbb{Z}$ as suitable inductive limits over ${\rm II}_1$-factors. We show that the Kasparov product, together with its usual functorial properties, extends to $KK$-theory with real coefficients. Let $Γ$ be a group. We define a $Γ$-algebra $A$ to be $K$-theoretically free and proper (KFP) if the group trace ${\bf tr}$ of $Γ$ acts as the unit element in $KK^Γ_{\mathbb{R}}(A,A)$. We show that free and proper $Γ$-algebras (in the sense of Kasparov) have the (KFP) property. Moreover, if $Γ$ is torsion free and …
Let X be a locally compact space, and let A and B be Co(X)-algebras. We define the notion of an asymptotic Co(X)-morphism from A to B and construct representable E-theory groups RE(X;A,B). These are the universal groups on the category of separable Co(X)-algebras that are Co(X)-stable, Co(X)-homotopy-invariant, and half-exact. If A is RKK(X)-nuclear, these groups are naturally isomorphic to Kasparov's representable KK-theory groups RKK(X;A,B). Applications and examples are also discussed.
Let $A$ be a graded C*-algebra. We characterize Kasparov's K-theory group $\hat{K}_0(A)$ in terms of graded *-homomorphisms by proving a general converse to the functional calculus theorem for self-adjoint regular operators on graded Hilbert modules. An application to the index theory of elliptic differential operators on smooth closed manifolds and asymptotic morphisms is discussed.
We study the noncommutative topology of the $C^*$-algebras $C(\mathbb{C}P_q^{n})$ of the quantum projective spaces within the framework of Kasparov's bivariant K-theory. In particular, we construct an explicit KK-equivalence with the commutative algebra $\mathbb{C}^{n+1}$. Our construction relies on showing that the extension of $C^*$-algebras relating two quantum projective spaces of successive dimensions admits a splitting, which we can describe explicitly using graph algebra techniques.
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.
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…
A group equivariant $KK$-theory for rings will be defined and studied in analogy to Kasparov's $KK$-theory for $C^*$-algebras. It is a kind of linearization of the category of rings by allowing addition of homomorphisms, imposing also homotopy invariance, invertibility of matrix corner embeddings, and allowing morphisms which are the opposite split of split exact sequences. We demonstrate the potential of this theory by proving for example equivalence induced by Morita equivalence and a Green-Julg isomorphism in this framework.
We consider Hilsum's notion of bordism as an equivalence relation on unbounded $KK$-cycles and study the equivalence classes. Upon fixing two $C^*$-algebras, and a $*$-subalgebra dense in the first $C^*$-algebra, a $\mathbb{Z}/2\mathbb{Z}$-graded abelian group is obtained; it maps to the Kasparov $KK$-group of the two $C^*$-algebras via the bounded transform. We study properties of this map both in general and in specific examples. In particular, it is an isomorphism if the first $C^*$-algebra is the complex numbers (i.e., for $K$-theory) and is a split surjection if the first $C^*$-algebra is…
The homotopy symmetric $C^*$-algebras are those separable $C^*$-algebras for which one can unsuspend in E-theory. We find a new simple condition that characterizes homotopy symmetric nuclear $C^*$-algebras and use it to show that the property of being homotopy symmetric passes to nuclear $C^*$-subalgebras and it has a number of other significant permanence properties. As an application, we show that if $I(G)$ is the kernel of the trivial representation $ι:C^*(G)\to \mathbb{C}$ for a countable discrete torsion free nilpotent group $G$, then $I(G)$ is homotopy symmetric and hence the Kasparov gr…
We develop methods for computing graded K-theory of C*-algebras as defined in terms of Kasparov theory. We establish graded versions of Pimsner's six-term sequences for graded Hilbert bimodules whose left action is injective and by compacts, and a graded Pimsner-Voiculescu sequence. We introduce the notion of a twisted P-graph C*-algebra and establish connections with graded C*-algebras. Specifically, we show how a functor from a P-graph into the group of order two determines a grading of the associated C*-algebra. We apply our graded version of Pimsner's exact sequence to compute the graded K…
Let $Γ$ be a lattice in a locally compact group $G$. In earlier work, we used $KK$-theory to equip the $K$-groups of any $Γ$-$C^{*}$-algebra on which the commensurator of $Γ$ acts with Hecke operators. When $Γ$ is arithmetic, this gives Hecke operators on the $K$-theory of certain $C^{*}$-algebras that are naturally associated with $Γ$. In this paper, we first study the topological $K$-theory of the arithmetic manifold associated to $Γ$. We prove that the Chern character commutes with Hecke operators. Afterwards, we show that the Shimura product of double cosets naturally corresponds to the Ka…
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…
The external Kasparov product is used to construct odd and even spectral triples on crossed products of $C^*$-algebras by actions of discrete groups which are equicontinuous in a natural sense. When the group in question is $\Z$ this gives another viewpoint on the spectral triples introduced by Belissard, Marcolli and Reihani. We investigate the properties of this construction and apply it to produce spectral triples on the Bunce-Deddens algebra arising from the odometer action on the Cantor set and some other crossed products of AF-algebras.
We analzye Rieffel's construction of generalized fixed point algebras in the setting of group actions on Hilbert modules. Let G be a locally compact group acting on a C*-algebra B. We construct a Hilbert module F over the reduced crossed product of G and B, using a pair (E, R), where E is an equivariant Hilbert module over B and R is a dense subspace of E with certain properties. The generalized fixed point algebra is the C*-algebra of compact operators on F. Any Hilbert module over the reduced crossed product arises by this construction for a pair (E, R) that is unique up to isomorphism. A ne…
Starting with a spectral triple on a unital $C^{*}$-algebra $A$ with an action of a discrete group $G$, if the action is uniformly bounded (in a Lipschitz sense) a spectral triple on the reduced crossed product $C^{*}$-algebra $A\rtimes_{r} G$ is constructed in [Hawkins, Skalski, White, Zacharias. Mathematica Scandinavica 2013]. The main instrument is the Kasparov external product. We note that this construction still works for twisted crossed products when the twisted action is uniformly bounded in the appropriate sense. Under suitable assumptions we discuss some basic properties of the resul…
Let A and B be separable nuclear continuous C(X)-algebras over a finite dimensional compact metrizable space X. It is shown that an element $σ$ of the parametrized Kasparov group KK_X(A,B) is invertible if and only if all its fiberwise components $σ_x\in KK(A(x),B(x))$ are invertible. This criterion does not extend to infinite dimensional spaces since there exist nontrivial unital separable continuous fields over the Hilbert cube with all fibers isomorphic to the Cuntz algebra O_2. Several applications to continuous fields of Kirchberg algebras are given. It is also shown that if each fiber of…
