|
In mathematics, KK-theory is a common generalization both of K-homology and K-theory (more precisely operator K-theory), as an additive bivariant functor on separable C *-algebras. This notion was introduced by the Russian mathematician Gennadi Kasparov〔G. Kasparov. The operator K-functor and extensions of C *-algebras. Izv. Akad. Nauk. SSSR Ser. Mat. 44 (1980), 571-636〕 in 1980. It was influenced by Atiyah's concept of Fredholm modules for the Atiyah–Singer index theorem, and the classification of extensions of C *-algebras by Brown–Douglas–Fillmore (, Ronald G. Douglas, Peter Arthur Fillmore 1977).〔Brown, L. G.; Douglas, R. G.; Fillmore, P. A., "Extensions of C *-algebras and K-homology", ''Annals of Mathematics'' (2) 105 (1977), no. 2, 265–324. 〕 In turn, it has had great success in operator algebraic formalism toward the index theory and the classification of nuclear C *-algebras, 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 actions. == 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 () 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〕 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 :, 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「KK-theory」の詳細全文を読む スポンサード リンク
|