In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, ''KK''-theory is a common generalization both of
K-homology and
K-theory
In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is a cohomology theory known as topological K-theory. In algebra and algebraic geometr ...
as an additive
bivariant functor on
separable C*-algebras
In mathematics, specifically in functional analysis, a C∗-algebra (pronounced "C-star") is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra ''A'' of continuous ...
. This notion was introduced by the Russian mathematician
Gennadi Kasparov in 1980.
It was influenced by Atiyah's concept of
Fredholm modules for the
Atiyah–Singer index theorem
In differential geometry, the Atiyah–Singer index theorem, proved by Michael Atiyah and Isadore Singer (1963), states that for an elliptic differential operator on a compact manifold, the analytical index (related to the dimension of the space ...
, and the classification of
extension
Extension, extend or extended may refer to:
Mathematics
Logic or set theory
* Axiom of extensionality
* Extensible cardinal
* Extension (model theory)
* Extension (proof theory)
* Extension (predicate logic), the set of tuples of values that ...
s of
C*-algebra
In mathematics, specifically in functional analysis, a C∗-algebra (pronounced "C-star") is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra ''A'' of contin ...
s by
Lawrence G. Brown,
Ronald G. Douglas, and Peter Arthur Fillmore in 1977. In turn, it has had great success in operator algebraic formalism toward the index theory and the classification of
nuclear C*-algebra
In the mathematical field of functional analysis, a nuclear C*-algebra is a C*-algebra such that for every C*-algebra the injective and projective C*- cross norms coincides on the algebraic tensor product and the completion of with respect ...
s, 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 plays a crucial role in
noncommutative topology.
''KK''-theory was followed by a series of similar bifunctor constructions such as the ''E''-theory and the
bivariant periodic cyclic theory, most of them having more
category-theoretic flavors, or concerning another class of algebras rather than that of the separable ''C''*-algebras, or incorporating
group action
In mathematics, a group action of a group G on a set S is a group homomorphism from G to some group (under function composition) of functions from S to itself. It is said that G acts on S.
Many sets of transformations form a group under ...
s.
Definition
The following definition is quite close to the one originally given by Kasparov. This is the form in which most KK-elements arise in applications.
Let ''A'' and ''B'' be separable ''C''*-algebras, where ''B'' is also assumed to be ''σ''-unital. The set of cycles is the set of triples , where ''H'' is a countably generated graded
Hilbert module over ''B'', ''ρ'' is a *-representation of ''A'' on ''H'' as even bounded operators that commute with ''B'', and ''F'' is a bounded operator on ''H'' of degree 1, which again commutes with ''B''. They are required to fulfill the condition that
:
for ''a'' in ''A'' are all ''B''-compact operators. 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 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.
The KK-group KK(''A'', ''B'') between ''A'' and ''B'' is then defined to be the set of cycles modulo homotopy. It becomes an abelian group under the direct sum operation of bimodules as the addition, and the class of the degenerate modules as its neutral element.
There are various, but equivalent definitions of the KK-theory, notably the one due to
Joachim Cuntz[J. Cuntz. A new look at KK-theory. K-Theory 1 (1987), 31–51] that eliminates bimodule and 'Fredholm' operator ''F'' from the picture and puts the accent entirely on the homomorphism ''ρ''. More precisely it can be defined as the set of homotopy classes
: