KK-theory

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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.

Contents 1 Definition 2 Examples 3 Properties 4 Bibliography

[edit] 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 σ − unital. The set of cycles is the set of triples , where is a countably generated graded Hilbert module over B, ρ is a *-homomorphism from A to the even bounded operators on , and D is an operator on of degree 1 such that

• [D,ρ(a)]
• (D² − 1)ρ(a)
• (D − D ^* )ρ(a)

are all compact operators for all .

(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 ρ'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 ρ. More precisely it can be defined as

,

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.

[edit] Examples

• by assigning to the operator its index.
• , B being a C* algebra.

[edit] Properties

The most important property is the so-called composition product

or, more generally,

,

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 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.

[edit] Bibliography

• B. Blackadar, Operator Algebra, Springer (2005)

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