The KK - theory is a mathematical theory in the field of functional analysis. The name comes from the fact that it is a K-theory with two variables, which generalizes the classical K-theory for C *-algebras and the theory of extensions of C * - algebras. The KK - theory goes back to G. G. Kasparov.

  • 2.1 Definition
  • 2.2 Tensor products of C *-algebras
  • 2.3 Bott periodicity
  • 2.4 Alternative description of KK1 (A, B)
  • 3.1 homomorphisms
  • 3.2 KK ( ℂ, ℂ )
  • 3.3 K- groups
  • 3.4 extensions
  • 4.1 in the first component functoriality
  • 4.2 functoriality in the second component
  • 5.1 Construction and properties
  • 5.2 Advanced Kasparov product


The groups to be defined below the KK - theory are formed by equivalence classes of Hilbert C * -modules with an additional structure and depend on two C *-algebras and down. The C *-algebras carry a graduation and for some part very technical constructions should all C *-algebras that occur as the first variable to be separable and all C *-algebras that occur as a second variable, σ - unital, which simplicity is always provided for below. For C *-algebras without graduation can be the trivial graduation, which is generated by the identity as Graduierungsautomorphismus insinuate; then obtained statements for non -graded C * - algebras.

Kasparov modules

A Kasparov AB- module is a triple consisting of

  • A countably generated graded Hilbert - Law Module,
  • A graduated * - homomorphism, where the C * - algebra of linear operators on the arising of graduation (see Hilbert C * - module ),
  • An operator with the following properties, Ie is homogeneous of degree 1,
  • For all,
  • For all,
  • For everyone.

It denotes the commutator graduate and the two-sided ideal of " compact " operators in. The letter for the operator to remember Fredholm operator. Note that it is carried to an A -B - bimodule.

Considering the elements of a " small" compared to the more general, so how sometimes perceives a real compact operator as a "small" perturbation of a Hilbert space operator, the conditions say that the sizes are supposed to be " small". Are these sizes even 0, it is called the Kasparov module degenerate. The trivial module is a degenerate Kasparov AB- module; Note that this very different objects are denoted by 0.

Below is the class of Kasparov AB -modules. We will define an addition and a suitable equivalent ratio, which makes the set of equivalence classes in a group. This will be the KK - groups.

Direct sum

The addition in the KK - groups will be defined by means of the direct sum, therefore, we briefly describe the direct sum of finitely many Kasparov modules. Be Kasparov A-B -modules. We define the direct sum


With graduation, with the graduations on the be.

One checks that the triple a Kasparov AB- module is again, which is called the direct sum of.


At the announced definition of the equivalence relation on we will need the following as a pushout designated construction. It is a Kasparov -AB module and a surjective * - homomorphism. Then one can show that for every operator exactly one operator returns with, the pushout of the Hilbert B- module with respect and the quotient map. Then defines a * - homomorphism and

Is a Kasparov -AC module, which is called the pushout of respect.

Equivalence relations

Similar to the K- theory in which the direct sum defines a shortcut on suitable equivalence classes, we will proceed here. It is common to define several equivalence relations, one of which is shown then that they attached to the C *-algebras, which we mentioned above generally require under appropriate Abzählbarkeitsvoraussetzungen coincide. We are only one of these equivalence relations, the so-called homotopy relation turn to us here and mention at this point that you developed parts of the theory in parallel for several equivalence relations to then later show equality can. Since we want to represent only results, we will not trace this path here.

To define the homotopy relation, we first need the finer unitary equivalence. Two Kasparov AB -modules and are called unitarily equivalent, in symbols, if there is an operator that gives a unitary equivalence of graded Hilbert C * -modules, so that

  • For all

For a graded C *-algebra is the C * - algebra of continuous functions on the unit interval with values ​​in. This is the graduation of induced again a graded C * - algebra. The evaluations are surjective * - homomorphisms, which therefore pushouts can be formed. Now you writes, if there is a Kasparov - AC ( [0,1], B) - module with and.

Finally, and called homotopic, in symbols, if there is finite amount of Kasparov AB -modules with

It shows that an equivalence relation on the class of Kasparov AB -modules is. The equivalence class of a Kasparov module is denoted by.

The KK - groups


There are and C *-algebras. Then be

The set of equivalence classes of Kasparov AB -modules. Since the countably generated Hilbert B -modules can be regarded as submodules of stabilization after the set of unitary equivalence up to Kasparov, here is actually a lot before. It can be shown that the sum of a direct addition

Defined, which makes it an abelian group.

The zero element is the class of degenerate Kasparov modules. We briefly describe the inverses in this group. It is

It is the Hilbert - module to the opposite graduation, i.e., the modules have the same homogeneous elements only the degree of the elements of a are homogeneous with respect to the elements of the other levels of the homogeneous shifted by 1 modulo. Further, the elements of the homogeneous of degree are.

Finally, it

Here, the tensor product is graded and, where these two-dimensional C * - algebra defined by the wearing graduation. To better understand this tensor product of Graduierungsautomorphismus was on and be the one homomorphism of 0 maps to the identity and 1 on. Then is a C * - dynamical system and the graduated tensor product is isomorphic to the cross product of this C * - dynamical system.

Tensor products of C *-algebras

There are a Kasparaow -AB module further, separable, graded C * - algebra. Then a graduate Hilbert D- module and you can the Hilbert B ⊗ D module, the outer tensor product of two graded Hilbert C * -modules form. * is then a homomorphism and an operator who makes a Kasparow A ⊗ ⊗ D DB module. In this way we obtain a homomorphism

Bott periodicity

We write the group as. By iterated with the formation of the graduate tensor, one could also define higher KK - groups:

But that turns out to be unnecessary, because one can show that

This is called the formal Bott periodicity, because it behaves similar to the Bott periodicity of K- theory. The formal Bott periodicity can be attributed primarily to the relationship and is therefore much simpler than the real Bott periodicity that uses mounts. But these real Bott periodicity can be proved in the KK - Theroie.

Is a C *-algebra, we denote by the device to attach from, that is, the C * - algebra of all continuous functions which vanish at infinity. Then we have

  • ,

Alternative description of KK1 (A, B)

Here, we define so-called CC1 -cycle and show how using an appropriate equivalence relation such Cycles to an alternative description of *-algebras can be used for trivial C graduated.

A CC1 -cycle is a pair consisting of an element and a * - homomorphism such that

It should be the amount of such KK1 -cycle. Two KK1 -cycle and hot homomtop, if there is to

Wherein the evaluation point in the Figure is unique and their strictly - continuous extension to the Multiplikatorenalgebren. This with designated relation is an equivalence relation whose equivalence classes we write with square brackets.

On one has by

Defined addition, a unitary isomorphism call.

We now describe according to an isomorphism of. In this regard it and by defined isomorphism. For a set KK1 -cycle

Then and

Is an isomorphism. This is described regardless of the formal Bott periodicity.



Is a graduate * - homomorphism between C *-algebras, so is a Kasparov AB- module. Note to the agreed above condition is that - unital; so as Hilbert - actually generated countable ( with trivial graduation) module. The equivalence class is often referred to with. The elements are to assume the role of identities in the below to be discussed Kasparov product.

KK ( ℂ, ℂ )

There were with with graduation and the * - homomorphism. Furthermore, the quotient map is in the Calkin algebra. Be an operator, so that unitary. Then

Since unitary is, is a Fredholm operator, and one can show that

A group isomorphism, where index denotes the Fredholm index.

K- groups

We show here how the K- groups of a C * - algebra appear ( with trivial graduation) in the KK - theory again.

Whether it is a unitary member of the outer multipliers Algebra Algebra of C *, that is, from the ratio of the multipliers algebra tensor and the C *-algebra operators of the compact to a separable Hilbert space for this tensor. Be a foot lifter of, that is. Then

Note here that, and therefore, the third component of the specified element is actually and clearly has the level 1. is the * - homomorphism. Then it can be shown that the association of

A group isomorphism is. As stated in the article about Multiplikatorenalgebren, one also has a natural isomorphism, so that one obtains a total isomorphism.

Either by similar considerations or by using the above presented Bott periodicity one also comes to a isomorphism, so that you get a total of slightly more memorable to the following formula:


Using the above- featured alternative description of means KK1 -cycles can be constructed an isomorphism, where the former denotes the group of invertiebaren elements in Ext ( A, B). As stated in the article on extensions of C * - algebras belong to the Busby invariant of an invertible element a homomorphism and a projection. Then a KK1 -cycle and we get a group isomorphism


The assignment of two C *-algebras to their KK - group can be extended to a functor when each fixed a C * - algebra. These functors are even shown as homotopieinvariant.

Functoriality in the first component

Are a Kasparov AB- module and a graded * - homomorphism, so is a Kasparov -CB module and gives a group homomorphism

This is for a fixed to a contravariant functor from the category of separable, graded C *-algebras in the category of abelian groups. Looking at each C * - algebra is the trivial graduation, we obtain a contravariant functor from the category of separable C *-algebras in the category of abelian groups.

This functor is homotopieinvariant, that is, are * - homomorphisms for, so that the pictures are continuous for all, then.

Functoriality in the second component

Are a Kasparov AB- module and a graded * - homomorphism, enlighten the inner tensor product. This is a Hilbert - module and is a * - Homomorphhismus. By this definition, we obtain a group homomorphism

This is for a fixed to a covariant functor from the category of - unital, graded C *-algebras in the category of abelian groups. Looking at each C * - algebra is the trivial graduation, we obtain a covariant functor from the category of unital C * - algebras in the category of abelian groups.

This functor is homotopieinvariant, that is, are * - homomorphisms for, so that the pictures are continuous for all, then.

The Kasparov product

Structure and properties

The Kasparov product is a mapping

That in the applications is a powerful tool. Both the structure which can only be indicated below, as well as the proof of the properties listed below require a high technical complexity.

And in no construction. Then this is graduated, inner tensor product is a Hilbert - module and is a * - homomorphism. With great technical effort one can construct a suitable operator and so define an element, which is called the Kasparaow - product of the two elements and, and the following properties:

  • For * - homomorphism
  • For * - homomorphism
  • For * - homomorphism
  • For * - homomorphism
  • For * - homomorphism.

In particular, a ring with one element for each separable C * algebra. The above- presented group isomorphism turns out to be ring isomorphism. Is an AF C * - algebra, it is isomorphic to the endomorphism ring of the group.

Advanced Kasparov product

The Kasparov product can be as follows to a product

Be generalized, where the C *-algebras arising to meet the above agreed Abzählbarkeitsbedingungen and the tensor product is always the graduated tensor. For and

So you can make the Kasparov product, an element in. This product is referred to again and confirmed that it does not conflict with the already defined Kasparov product, primarily because of the fact that the identity is. Overall, we obtain the announced, bilinear map

For you get back the already known Kasparov product, because that tensoring with leads to isomorphic C * - algebras and is the identity. In this sense, the above product is a generalization of the previously introduced Kasparov product dar.

As an important special case, we want to treat the tensoring with, because that leads to the above definition to KK1 groups. Is special and so can pass on the occurrence of the tensor product with the CC1 group and the tensoring with fortlassen. From the above, therefore, one obtains a bilinear map

So you can multiply elements from the left by elements and receives an item.

By setting analog and respectively, we obtain bilinear mappings


Where for the last figure is still the formal Bott periodicity was used.

Cyclic, 6 -membered, exact sequences

Known from K. theory cyclic 6- membered exact sequences can be proved in the small-bore theory. We go from a short exact sequence

From which arises from a closed, two-sided ideal in a C * - algebra with identity and of which we shall suppose that it is semi- divisive. Then this sequence is an invertible extension and therefore determined according to the above one element. The plane defined by the Kasparov product with multiplication defined for another C *-algebra homomorphisms

Which are referred to with all. Then there are the following cyclic exact sequences:


Such sequences are known in the calculation of KK - groups often helpful, particularly if one or two members of such a sequence are 0, because then some pictures using precision can be detected as injective, surjective or even bijective.