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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 : , \rho(a) (F^2-1)\rho(a), (F-F^*)\rho(a) 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 CuntzJ. 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 : KK(A,B) = A, K(H) \otimes B/math>, of *-homomorphisms from the classifying algebra ''qA'' of quasi-homomorphisms to the ''C''*-algebra of compact operators of an infinite dimensional separable Hilbert space 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.


Properties

When one takes the ''C''*-algebra C of the complex numbers as the first argument of ''KK'' as in ''KK''(C, ''B'') this additive group is naturally isomorphic to the ''K''0-group ''K''0(''B'') of the second argument ''B''. In the Cuntz point of view, a ''K''0-class of ''B'' is nothing but a homotopy class of *-homomorphisms from the complex numbers to the stabilization of ''B''. Similarly when one takes the algebra ''C''0(R) of the continuous functions on the real line decaying at infinity as the first argument, the obtained group is naturally
isomorphic In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
to ''K''1(''B''). An important property of ''KK''-theory is the so-called Kasparov product, or the composition product, : KK(A,B) \times KK(B,C) \to KK(A,C), which is bilinear with respect to the additive group structures. In particular each element of gives a homomorphism of and another homomorphism . 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 that induce ''KK''-equivalences. The composition product gives a new
category Category, plural categories, may refer to: General uses *Classification, the general act of allocating things to classes/categories Philosophy * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) * Category ( ...
\mathsf, whose objects are given by the separable ''C''*-algebras while the morphisms between them are given by elements of the corresponding KK-groups. Moreover, any *-homomorphism of ''A'' into ''B'' induces an element of and this correspondence gives a functor from the original category of the separable ''C''*-algebras into \mathsf. The approximately inner automorphisms of the algebras become identity morphisms in \mathsf. This functor \mathsf \to \mathsf is universal among the split-exact, homotopy invariant and stable additive functors on the category of the separable ''C''*-algebras. Any such theory satisfies
Bott periodicity In mathematics, the Bott periodicity theorem describes a periodicity in the homotopy groups of classical groups, discovered by , which proved to be of foundational significance for much further research, in particular in K-theory of stable compl ...
in the appropriate sense since \mathsf does. The Kasparov product can be further generalized to the following form: : KK(A, B \otimes E) \times KK(B \otimes D, C) \to KK(A \otimes D, C \otimes E). It contains as special cases not only the K-theoretic
cup product In mathematics, specifically in algebraic topology, the cup product is a method of adjoining two cocycles of degree p and q to form a composite cocycle of degree p+q. This defines an associative (and distributive) graded commutative product opera ...
, but also the K-theoretic
cap A cap is a flat headgear, usually with a visor. Caps have crowns that fit very close to the head. They made their first appearance as early as 3200 BC. The origin of the word "cap" comes from the Old French word "chapeau" which means "head co ...
, cross, and slant products and the product of extensions.


Notes


References

* B. Blackadar
''Operator Algebras: Theory of C*-Algebras and Von Neumann Algebras''
Encyclopaedia of Mathematical Sciences 122, Springer (2005) * A. Connes, ''Noncommutative Geometry'', Academic Press (1994)


External links

* * {{nlab, id=E-theory K-theory C*-algebras