KK-theory

:"This article is on the generalization of operator K-theory and K-homology. For the epistemological concept, see KK-principle."

In mathematics, KK-theory is a common generalization both of topological K-homology and K-theory (more precisely operator K-theory), by means of a bivariant functor on separable C*-algebras, defined from 1981 on by Gennadi Kasparov.

It has significant input from and relations with index theory, elliptic operator theory, and classification of extensions of C*-algebras.

It has had great success in the field and influenced it significantly, as it was the key to the solutions of many problems in operator K-theory, such as, for instance, the mere calculation of K-groups.

Furthermore, it was essential in the development of the Baum-Connes conjecture and is crucial in noncommutative topology.

Definition

The following definition is quite close to Kasparov's original one. This is the form in which most KK-elements arise in applications.

Let A and B be separable C*-algebras, where B is $sigma-unital$. The set of cycles is the set of triples $(mathcal\{H\},\; ho,D)$, where $mathcal\{H\}$ is a countably generated graded Hilbert module over $B$, $ho$ is a *-homomorphism from $A$ to the even bounded operators on $mathcal\{H\}$, and $D$ is an operator on $mathcal\{H\}$ of degree 1 such that
* $[D,\; ho(a)]$
* $(D^2-1)\; ho(a)$
* $(D-D^*)\; ho(a)$are all compact operators for all $ain\; A$.

(A cycle is said to be degenerate if all three expressions are 0 for all $a$.)

Two cycles are said to be homologous, or homotopic, if there is a cycle between $A$ and $IB$, where $IB$ denotes the C*-algebra of continuous functions from $[0,1]$ to $B$, such that there is an even unitary operator from the $0$-end of the homotopy to the first cycle, and a unitary operator from the $1$-end of the homotopy to the second cycle, each compatible with the $ho$'s and the $D$'s.

$KK(A,B)$ is then defined to be the set of cycles modulo homotopy.

There are various different, but equivalent definitions of KK-theory, notably one due to Joachim Cuntz which eliminates the operator from the picture and puts the accent entirely on the homomorphism $ho$. More precisely it can be defined as

$KK\_\{\; ext\{Cuntz(A,B)=\; [qA,mathcal\{K\}otimes\; B]$,

the set of homotopy classes of *-homomorphisms from the classifying algebra $qA$ of quasi-homomorphisms to the C*-algebra of compact operators tensored with $B$. Here, $qA$ is defined as the kernel of the map from the C*-algebraic free product $A*A$ of $A$ with itself to $A$ defined by the identity on both factors.

Example

$KK(mathbb\{C\},mathbb\{C\})=mathbb\{Z\}$ by assigning to the operator its index.

Properties

The most important property is the so-called composition product

$KK(A,B)\; imes\; KK(B,C)\; ightarrow\; KK(A,C)$

or, more generally,

$KK(A,Botimes\; E)\; imes\; KK(Botimes\; D,C)\; ightarrow\; KK(Aotimes\; D,Cotimes\; E)$,

of which the ring structure in ordinary K-theory is a special case. It contains as special cases not only the K-theoretic cup product, but also the K-theoretic cap, cross, and slant products and the product of extensions.

The product can be defined much more easily in the Cuntz picture given that there are natural maps from $qA$ to $A$, and from $B$ to $mathcal\{K\}otimes\; B$ which induce KK-equivalences.

The product gives the structure of a category to KK.

Another important property of KK-theory is that it is the universal split-exact, homotopy invariant and stable bifunctor on separable C*-algebras. Any such theory satisfies Bott periodicity in the appropriate sense.