In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, ''KK''-theory is a common generalization both of
K-homology In mathematics, K-homology is a homology theory on the category of locally compact Hausdorff spaces. It classifies the elliptic pseudo-differential operators acting on the vector bundles over a space. In terms of C^*-algebras, it classifies the F ...
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 module In noncommutative geometry, a Fredholm module is a mathematical structure used to quantize the differential calculus. Such a module is, up to trivial changes, the same as the abstract elliptic operator introduced by .
Definition
If ''A'' is an in ...
s for the
Atiyah–Singer index theorem, 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 (predicate logic), the set of tuples of values that satisfy the predicate
* E ...
s of
C*-algebras by
Lawrence G. Brown,
Ronald G. Douglas
Ronald George Douglas (December 10, 1938 – February 27, 2018) was an American mathematician, best known for his work on operator theory and operator algebras.
Education and career
Douglas was born in Osgood, Indiana. He was an undergraduate a ...
, 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 A such that the injective and projective C*- cross norms on A \oplus B are the same for every C*-algebra B. This property was first studied by under the name " ...
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 In mathematics, specifically in operator K-theory, the Baum–Connes conjecture suggests a link between the operator K-theory, K-theory of the reduced C*-algebra of a group theory, group and the K-homology of the classifying space of proper act ...
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 on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism ...
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 (''H'', ρ, ''F''), where ''H'' is a countably generated graded
Hilbert module over ''B'', ρ is a *-representation of ''A'' on ''H'' as even bounded operators which 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
,1to ''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] which 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
: