## PAPERS

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C*-algebraic higher signatures and an invariance theorem in codimension two

(With Thomas Schick and Zhizhang Xie.) We revisit the construction of signature classes in C*-algebra K-theory, and develop a variation that allows us to prove equality of signature classes in some situations involving homotopy equivalences of noncompact manifolds that are only defined outside of a compact set. As an application, we prove a counterpart for signature classes of a codimension two vanishing theorem for the index of the Dirac operator on spin manifolds (the latter is due to Hanke, Pape and Schick, and is a development of well-known work of Gromov and Lawson).

Algebraic families of groups and commuting involutions

(With Dan Barbasch and Eyal Subag.) Let G be a complex affine algebraic group equipped with two commuting, anti-holomorphic involutions. We construct an algebraic family of algebraic groups over the complex projective line and a real structure on the family that interpolates between the real forms associated to the two involutions.

Contractions of representations and algebraic families of Harish-Chandra modules

(With Joseph Bernstein and Eyal Subag.) We examine from an algebraic point of view some families of unitary group representations that arise in mathematical physics and are associated to contraction families of Lie groups. The contraction families of groups relate different real forms of a reductive group and are continuously parametrized, but the unitary representations are defined over a parameter subspace that includes both discrete and continuous parts. Both finite- and infinite-dimensional representations can occur, even within the same family. We shall study the simplest nontrivial examples, and use the concepts of algebraic families of Harish-Chandra pairs and Harish-Chandra modules, introduced in a previous paper, together with the Jantzen filtration, to construct these families of unitary representations algebraically.

Euler-like vector fields and manifolds with filtered structure

(With Ahmad Reza Haj Saeedi Sadegh.) The first purpose of this note is to comment on a recent article of Bursztyn, Lima and Meinrenken, in which it is proved that if M is a smooth submanifold of a manifold V, then there is a bijection between germs of tubular neighborhoods of M and germs of "Euler-like" vector fields on V. We shall explain how to approach this bijection through the deformation to the normal cone that is associated to the embedding of M into V. The second purpose is to study generalizations to smooth manifolds equipped with Lie filtrations. Following in the footsteps of several others, we shall define a deformation to the normal cone that is appropriate to this context, and relate it to Euler-like vector fields and tubular neighborhood embeddings.

On a spectral theorem of Weyl

(With Qijun Tan.) We give a geometric proof of a theorem of Weyl on the continuous part of the spectrum of Sturm-Liouville operators on the half-line with asymptotically constant coefficients. Earlier proofs due to Weyl and Kodaira depend on special features of Green's functions for linear ordinary differential operators; ours might offer better prospects for generalization to higher dimensions, as required for example in noncommutative harmonic analysis.

A differential complex for CAT(0) cubical spaces

(With Jacek Brodzki and Erik Guentner.) In the 1980's Pierre Julg and Alain Valette, and also Tadeusz Pytlik and Ryszard Szwarc, constructed and studied a certain Fredholm operator associated to a simplicial tree. The operator can be defined in at least two ways: from a combinatorial flow on the tree, similar to the flows in Forman's discrete Morse theory, or from the theory of unitary operator-valued coccyges. There are applications of the theory surrounding the operator to C*-algebra K-theory, to the theory of completely bounded representations of groups that act on trees, and to the Selberg principle in the representation theory of p-adic groups. The main aim of this paper is to extend the constructions of Julg and Valette, and Pytlik and Szwarc, to CAT(0) cubical spaces. A secondary aim is to illustrate the utility of the extended construction by developing an application to operator K-theory and giving a new proof of K-amenability for groups that act properly on bounded-geometry CAT(0)-cubical spaces.

Algebraic families of Harish-Chandra pairs

(With Joseph Bernstein and Eyal Subag.) Mathematical physicists have studied degenerations of Lie groups and their representations, which they call contractions. In this paper we study these contractions, and also other families, within the framework of algebraic families of Harish-Chandra modules. We construct a family that incorporates both a real reductive group and its compact form, separate parts of which have been studied individually as contractions. We give a complete classification of generically irreducible families of Harish-Chandra modules in the case of the family associated to SL(2,R).

A second adjoint theorem for SL(2,R)

(With Tyrone Crisp.) We formulate a second adjoint theorem in the context of tempered representations of real reductive groups, and prove it in the case of SL(2,R).

Parabolic induction, categories of representations and operator spaces

(With Tyrone Crisp.) We study some aspects of the functor of parabolic induction within the context of reduced group C*-algebras and related operator algebras. We explain how Frobenius reciprocity fits naturally within the context of operator modules, and examine the prospects for an operator algebraic formulation of Bernstein's reciprocity theorem (his second adjoint theorem).

Adjoint functors between categories of Hilbert C*-modules

(With Pierre Clare and Tyrone Crisp.) Let E be a (right) Hilbert C*-module over a C*-algebra A. If E is equipped with a left action of a second C*-algebra B, then tensor product with E gives rise to a functor from the category of Hilbert B-modules to the category of Hilbert A-modules. The purpose of this paper is to study adjunctions between functors of this sort. We shall introduce a new kind of adjunction relation, called a local adjunction, that is weaker than the standard concept from category theory. We shall give several examples, the most important of which is the functor of parabolic induction in the tempered representation theory of real reductive groups. Each local adjunction gives rise to an ordinary adjunction of functors between categories of Hilbert space representations. In this way we shall show that the parabolic induction functor has a simultaneous left and right adjoint, namely the parabolic restriction functor constructed in a previous paper.

Parabolic induction and restriction via C*-algebras and Hilbert C*-modules

(With Pierre Clare and Tyrone Crisp.) This paper is about the reduced group C*-algebras of real reductive groups, and about Hilbert C*-modules over these C*-algebras. We shall do three things. First we shall apply theorems from the tempered representation theory of reductive groups to determine the structure of the reduced C*-algebra (the result has been known for some time, but it is difficult to assemble a full treatment from the existing literature). Second, we shall use the structure of the reduced C*-algebra to determine the structure of the Hilbert C*-bimodule that represents the functor of parabolic induction. Third, we shall prove that the parabolic induction bimodule admits a secondary inner product, using which we can define a functor of parabolic restriction in tempered representation theory.

K-theory and the quantization commutes with reduction problem

(With Yanli Song.) The purpose of this paper is to examine the quantization commutes with reduction phenomenon from the point of view of topological K-theory and K-homology. Our new contribution is an explanation, in the later sections of the paper, of the role Bott periodicity and the Weyl character formula in the transition from the commutative to the noncommutative cases of the quantization commutes with reduction problem.

Weyl character formula in KK-theory

(With Jonathan Block.) 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 KK-theory. Then we shall show how, contingent on the Baum-Connes conjecture, our KK-theoretic Weyl character formula can be carried over to noncompact groups.

On the analogy between complex semisimple groups and their Cartan motion groups

In a 1975 article George Mackey examined analogies between the representations of a semisimple Lie group and those of its Cartan motion group. Alain Connes later pointed out that the analogies observed by Mackey harmonize very well with the Connes-Kasparov conjecture in C*-algebra K-theory. Motivated by Connes' observation, I analyzed the reduced C*-algebra of a complex semisimple group in a way that led to a new confirmation of the Connes-Kasparov conjecture for such groups, while at the same time exhibiting a natural bijection between the tempered duals of the group and its Cartan motion group. The purpose of this article is examine the same issues algebraically rather than C*-algebraically, and so obtain a Mackey-type bijection between the admissible dual of a complex semisimple group and that of its motion group.

A geometric description of equivariant K-homology for proper actions

(With Paul Baum and Thomas Schick.) Let G be a discrete group and let X be a G-finite, proper G-CW-complex. We prove that the Kasparov equivariant K-homology groups of X are isomorphic to the geometric equivariant K-homology groups that are obtained by making the geometric K-homology theory of Baum and Douglas equivariant in the natural way. This reconciles the original and current formulations of the Baum-Connes conjecture for discrete groups.

The tangent groupoid and the index theorem

We present a proof of the index theorem for Dirac operators that is drawn from Connes' tangent groupoid approach, as described in his book Noncommutative Geometry.

K-homology, assembly and rigidity theorems for relative eta invariants

(With John Roe.) We connect the assembly map in C*-algebra K-theory to rigidity properties for relative eta invariants that have been investigated by Mathai, Keswani, Weinberger and others. We give a new and conceptual proof of Keswani's theorem that whenever the C*-algebra assembly map is an isomorphism, the relative eta invariants associated to the signature operator are homotopy invariants, whereas the relative eta invariants associated to the Dirac operator on a manifold with positive scalar curvature vanish.

The Mackey analogy and K-theory

In an interesting article from the 1970's, Mackey made a proposal to study representation theory for a semisimple group G by developing an analogy between G and an associated semidirect product group. We shall examine the connection between Mackey's proposal and C*-algebra K-theory. In one direction, C*-algebra theory prompts us to search for precise expressions of Mackey's analogy. In the reverse direction, we shall use Mackey's point of view to give a new proof of the complex semisimple case of the Connes-Kasparov conjecture in C*-algebra K-theory.

Weak amenability of CAT(0) cubical groups

(With Erik Guentner.) We prove that if G is a discrete group that admits a metrically proper action on a finite-dimensional CAT(0) cube complex, then G is weakly amenable.

On the equivalence of geometric and analytic K-homology

(With Paul Baum and Thomas Schick.) We give a proof that the geometric K-homology theory for finite CW-complexes defined by Baum and Douglas is isomorphic to Kasparov's K-homology. The proof is a simplification of more elaborate arguments which deal with the geometric formulation of equivariant K-homology theory.

The Atiyah-Singer index theorem

(With John Roe.) This is a short and informal introduction to the Atiyah-Singer index theorem for the Princeton Companion to Mathematics.

Operator algebras

(With John Roe.) This is a short and informal introduction to the theory of operator algebras for the Princeton Companion to Mathematics.

The Novikov conjecture for linear groups

(With Erik Guentner and Shmuel Weinberger.) Let K be a field. We show that every countable subgroup of GL(n, K) is uniformly embeddable in a Hilbert space. This implies that Novikov's higher signature conjecture holds for these groups. We also show that every countable subgroup of GL(2, K) admits a proper, affine isometric action on a Hilbert space. This implies that the Baum-Connes conjecture holds for these groups. Finally, we show that every subgroup of GL(n, K) is exact, in the sense of C*-algebra theory.

The residue index theorem of Connes and Moscovici

The main purpose of these notes is to present, in a self-contained way, a new and perhaps more accessible proof of the local index formula of Connes and Moscovici. But for the benefit of those who are just becoming acquainted with Connes' noncommutative geometry, we have also tried to provide some context for the formula by reviewing at the beginning of the notes some antecedent ideas in cyclic and Hochschild cohomology.

Mapping surgery to analysis I: Analytic signatures

(With John Roe.) This is the first of three articles which will explore in some detail the relationship between the K-theoretic higher index of the signature operator and the L-theoretic signature of M, around which surgery theory is organized and which is the usual context for discussions of manifolds and their signatures.

Mapping surgery to analysis II: Geometric signatures

(With John Roe.) This is the second of a series of three papers whose objective is to describe a C*-algebraic counterpart to the surgery exact sequence of Browder, Novikov, Sullivan and Wall. In the first paper, we defined an analytic signature invariant in C*-algebra K-theory. Such an invariant is associated to any analytically controlled Hilbert-Poincare complex, and it has homotopy invariance and bordism invariance properties in this analytic context. In this second paper we will show that analytically controlled Hilbert-Poincare complexes arise naturally from various geometric constructions.

Mapping surgery to analysis III: Exact sequences

(With John Roe.) This is the final paper in a series of three whose objective is to construct a natural transformation from the surgery exact sequence of Browder, Novikov, Sullivan and Wall to a long exact sequence of K-theory groups associated to a certain C*-algebra extension. In this paper we will complete the construction and it will turn out that the relationship between signatures and signature operators is fundamental to this construction. Briefly, to detect whether a homotopy equivalence of manifolds is a diffeomorphism, we may examine the mapping cylinder and ask whether this Poincare space (with boundary) is in fact a manifold (with boundary). In turn, this question may be addressed analytically by asking whether a suitable analytic signature associated to the Poincare space is actually the analytic index of some abstract elliptic operator.

The local index formula in noncommutative geometry

These notes provide an introduction to the local index formula of Connes and Moscovici. They emphasize the basic, analytic aspects of the subject. This is in part because the analysis must be dealt with first, before more purely cohomological issues are tackled, and in part because the later issues are already quite well covered in survey articles by Connes and others. Moreover, on the cohomological side, the final and definitive results have yet to be thoroughly investigated. I hope that the reader will be able to use these notes to introduce himself to these issues of current research interest.

Meromorphic continuation of zeta functions associated to elliptic operators

We give a new proof of the meromorphic continuation property of zeta functions associated to elliptic operators. The argument adapts easily to the hypoelliptic operators considered by Connes and Moscovici in their recent work on transverse index theory in noncommutative geometry.

Group C*-algebras and K-theory

(With Erik Guentner.) These notes are about the formulation of the Baum-Connes conjecture in operator algebra theory and the proofs of some cases of it. They are aimed at readers who have some prior familiarity with K-theory for -algebras (up to and including the Bott Periodicity theorem). The lectures begin by reviewing K-theory and the Bott periodicity theorem. Much of the Baum-Connes theory has to do with broadening the periodicity theorem in one way or another, and for this reason quite some time is spent formulating and proving the theorem in a way which is suited to later extensions. Following that, the lectures turn to the machinery of bivariant -theory and the formulation of the Baum-Connes conjecture. The main objective of the notes is reached in Lecture 4, where the conjecture is proved for groups which act properly and isometrically on affine Euclidean spaces. The remaining lectures deal with partial results which are important in applications and with counterxamples to various overly optimistic strengthenings of the conjecture.

Counterexamples to the Baum-Connes conjecture

(With Vincent Lafforgue and Georges Skandalis.) The purpose of this note is to present counterexamples to: the injectivity and the surjectivity of the Baum-Connes map for Hausdorff groupoids; the injectivity and surjectivity of the Baum-Connes map for (non-Hausdorff) holonomy groupoids of foliations; the surjectivity of the Baum-Connes map for coarse geometric spaces; and the surjectivity of the Baum-Connes map for discrete group actions on commutative C*-algebras, contingent on certain as yet unpublished results of Gromov.

Spaces with vanishing l2-homology and their fundamental groups

(With John Roe and Thomas Schick.) The purpose of this note is to prove the following results: (1) Let G be a finitely presented group and suppose that the homology of G with coefficients in l2(G) is zero in degrees 0,1 and 2. Then there is a connected 3-dimensional finite CW-complex X with fundamental group G such that the homology of G with coefficients in l2(G) is zero in all degrees. (2) Let G be a finitely presented group and suppose that the homology G with coefficients in l2(G) is zero in degrees 0,1 and 2. For every dimension n at least 6 there is a closed manifold M of dimension n and with fundamental group G such that the homology of G with coefficients in l2(G) is zero in all degrees.

E-theory and KK-theory for groups which act properly and isometrically on Hilbert space

(With Gennadi Kasparov.) The purpose of this article is to prove the Baum-Connes conjecture for an interesting and fairly broad class of groups, called by Gromov a-T-menable, and known to harmonic analysts as groups with the Haagerup approximation property.

Representations of p-adic groups: a view from operator algebras algebras

(With Paul Baum and Roger Plymen.) The purpose of these notes is to convey to a reasonably broad audience some byproducts of the authors' research into the C*-algebra K-theory of the p-adic group GL(N), which culminated in a proof of the Baum-Connes Conjecture in this case. Along the way to the proof a number of interesting issues came to light which we feel deserve some exposure, even though our understanding of them is far from complete, and is indeed mostly very tentative.

Bivariant K-theory and the Novikov conjecture

Kasparov's bivariant K-theory is used to prove two theorems concerning the Novikov higher signature conjecture. The first generalizes a result of J. Roe and the author on amenable group actions. The second is a C*-algebraic counterpart of a theorem of G. Carlsson and E. Pedersen.

Amenable group actions and the Novikov conjecture

(With John Roe.) Guoliang Yu has introduced a property of discrete metric spaces which guarantees the existence of a uniform embedding into Hilbert space. We show that the metric space underlying a finitely generated discrete group has this property if and only if the action of the group on its Stone-Cech compactification is topologically amenable. It follows from Yu's work that if BG is a finite complex, and if G acts amenably on some compact Hausdorff space, then the Novikov higher signature conjecture is true for G.

Equivariant E-theory for C*-algebras

(With Erik Guentner and Jody Trout.) The purpose of this article is to develop in some detail the theory of equivariant asymptotic morphisms, appropriate to C*-algebras equipped with continuous actions of locally compact groups, and so construct tools very similar to those of Kasparov's equivariant KK-theory for calculating the K-theory of group C*-algebras. A central problem in C*-algebra K-theory is the Baum-Connes conjecture, which proposes a formula for the K-theory of group C*-algebras. A primary goal of the paper is to first formulate the conjecture in the language of asymptotic morphisms, and then describe a general method, due essentially to Kasparov, for attacking various cases of it. At present the method encompasses nearly all that is known about the Baum-Connes conjecture.

The Baum-Connes conjecture

This report is a short account of past and recent work on a conjecture of P. Baum and A. Connes about the K-theory of group C*-algebras.

A Bott periodicity theorem for infinite-dimensional Euclidean space

(With Gennadi Kasparov and Jody Trout.) We formulate and prove an equivariant Bott periodicity theorem for infinite dimensional Euclidean vector spaces. The main features of our argument are (i) the construction of a non-commutative C*-algebra to play the role of the algebra of functions on infinite dimensional Euclidean space; and (ii) the construction of a certain index one elliptic partial differential operator which provides the basis for an inverse to the Bott periodicity map. These tools have applica- tions to index theory and the Novikov conjecture, notably a proof of the Novikov conjecture for amenable groups (the applications will be considered elsewhere).

Operator K-theory for groups which act properly and isometrically on Euclidean space

(With Gennadi Kasparov.) Let G be a countable discrete group which acts isometrically and metrically properly on an infinite-dimensional Euclidean space. We calculate the K-theory groups of the full and reduced C*-algebras of G. Our result is in accordance with the Baum-Connes conjecture.

A proof of the Baum-Connes conjecture for p-adic GL(n)

(With Paul Baum and Roger Plymen.) We give a proof of the Baum-Connes conjecture for p-adic GL(n).

C*-algebras and controlled topology

(With Erik Pedersen and John Roe.) This paper is an attempt to explain some aspects of the relationship between the K-theory of C*-algebras, on the one hand, and the categories of modules that have been developed to systematize the algebraic aspects of controlled topology, on the other. It has recently become apparent that there is a substantial conceptual overlap between the two theories, and this allows both the recognition of common techniques, and the possibility of new methods in one theory suggested by those of the other. We will define C*-algebras associated to various kinds of controlled structure and giving methods whereby their K-theory groups may be calculated in a number of cases.

Cyclic homology of totally disconnected groups acting on buildings

(With Victor Nistor.) We prove a homological counterpart of a conjecture of P. Baum and A. Connes concerning K-theory for convolution C*-algebras of p-adic groups by calculating the periodic cyclic homology for the convolution algebra of a totally disconnected group acting properly on a building.

On the coarse Baum-Connes conjecture

(With John Roe.) The purpose of this paper is to give a precise formulation of the Baum-Connes conjecture in coarse geometry and to prove the conjecture for spaces which are non-positively curved in some sense, including affine buildings and hyperbolic metric spaces in the sense of Gromov.

A homotopy invariance theorem in coarse cohomology and K-theory

(With John Roe.) We introduce a notion of homotopy which is appropriate to the coarse geometry and topology studied by the second author. We prove the homotopy invariance of coarse cohomology, and of the K-theory of the C*-algebra associated to a coarse structure on a space. We apply our homotopy invariance results to show that if M is a Hadamard manifold then the inverse of the exponential map at any point 0 induces an isomorphism between the K-theory groups of the C*-algebras associated to M and its tangent space at 0. This result is consistent with a coarse version of the Baum-Connes conjecture.

A note on Toeplitz operators

(With Erik Guentner.) We study Toeplitz operators on Bergman spaces using techniques from the analysis of Dirac-type operators on complete Riemannian manifolds, and prove an index theorem of Boutet de Monvel from this point of view.

Classifying space for proper actions and K-theory of group C*-algebras

(With Paul Baum and Alain Connes.) We announce a reformulation of the conjecture in C*-algebra K-theory formulated by the first two authors. The advantage of the new version is that it is simpler and applies more generally than the earlier statement. A key point is to use the universal example for proper actions.

Equivariant homology for SL(2) of a p-adic field

(With Paul Baum and Roger Plymen.) Let F be a p-adic field and let G be the group of unimodular 2 by 2 matrices over F . The aim of this paper is to calculate certain equivariant homology groups attached to the action of G on its tree. They arise in connection with a theorem of M. Pimsner on the K-theory of the C*-algebra of G, and our purpose is to explore the representation theoretic content of Pimsner's result.

On the K-theory proof of the index theorem

This paper is an exposition of the K-theory proof of the Atiyah-Singer Index Theorem. I have tried to separate, as much as possible, the analytic parts of the proof from the topological calculations. For the topology I have taken advantage of the Chern isomorphism to work mostly within the world of ordinary cohomology. The analytic part of the proof is done within the framework of asymptotic morphisms.

A coarse Mayer-Vietoris principle

(With John Roe and Guoliang Yu.) In this note we will show that for suitable decompositions of a metric space there are Mayer-Vietoris sequences both in coarse cohomology and in the K-theory of the C*-algebra. As an application we shall calculate the K-theory of the C*-algebra associated to a metric cone. The result is consistent with the calculation of the coarse cohomology of the cone, and with a 'coarse' version of the Baum-Connes conjecture.

The Weyl-von Neumann theorem for multipliers of some AF-algebras

(With Mikael Rordam.) A well-known theorem theorem of Weyl and von Neumann asserts that if X is a self-adjoint operator on a separable Hilbert space, then X is unitarily equivalent to a diagonal operator, modulo compact operators. In this paper we shall prove a result about self-adjoint elements in the multiplier algebra of a simple AF algebra I with unique trace which reduces to the Weyl-von Neumann theorem in the case where I is the C*-algebra of compact operators.

A note on the cobordism invariance of the index

The purpose of this note is to give a simple proof of the cobordism invariance of the analytic index of Dirac-type operators.

Deformations, morphismes asymtotiques et K-theorie bivariante

(With Alain Connes.) We show that stable homotopy classes of asymptotic morphisms from A to B, where A and B are C*-algebras, form an abelian group E(A,B), and that the corresponding bivariant functor is the universal half-exact functor whose abstract existence was shown by the second author. The theory thus obtained is an improvement and simplification of the bivariant theory KK(A,B) of Kasparov.

C*-algebra extension theory and duality

We develop a duality theory introduced by W. Paschke to give a simplified account of the main results of the Brown Douglas Fillmore extension theory and the relative K-homology theory of Baum and Douglas.

A primer on KK-theory

The purpose of this article is to acquaint the reader with Kasparov's KK-theory.

Categories of fractions and excision in KK-theory

Using elementary ideas from the theory of categories of fractions, we construct bivariant homology/cohomology groups E(A,B) for C*-algebras which satisfy general excision axioms and are equal to Kasparov's groups KK(A,B) for nuclear (or more generally K-nuclear) C*-algebras.

An approach to Z/k-index theory

This paper gives a C*-algebra K-theory proof of the index theorem for Z/k-manifolds due to Freed and Melrose.

Algebraic K-theory of stable C*-algebras

Let A be a unital C*-algebra and let Q denote the Calkin algebra of bounded operators on a separable Hilbert space modulo compact operators. We prove the following conjecture of Max Karoubi: the algebraic and topological K-theory groups of the C*-algebra tensor product of A and Q are equal.

On a technical theorem of Kasparov

This paper gives a short proof of the main technical theorem used by Kasparov to construct the product in KK-theory.

Kuiper's theorem for Hilbert modules

(With Joachim Cuntz.) The purpose of this note is to give a short proof that the unitary group of the mulitiplier algebra of a stable C*-algebra with a countable approximate unit is contractible in the norm topology.

A characterization of KK-theory

We characterize the KK-groups of G.G. Kasparov, along with the Kasparov product KK(A,B) x KK(B,C) -> KK(A,C), from the point of view of category theory (in a very elementary sense): the product is regarded as a law of composition in a category and we show that this category is the universal one with homotopy invariance, stability, and split exactness. The third property is a weakened type of half-exactness: it amounts to the fact that the KK-groups transform split exact sequences of C*-algebras to split exact sequences of abelian groups. The method is borrowed from Joachim Cuntz's apporach to KK-theory, in which cycles for KK(A,B) are regarded as generalized homomorphisms from A to B: the results follow from an analysis of the Kasparov product in this light.