KK-theory
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| KK-theory | |
|---|---|
| Name | KK-theory |
| Field | Noncommutative topology; Operator algebras |
| Introduced | 1980 |
| Inventor | Gennadi Kasparov |
| Notable examples | C*-algebras; Group C*-algebras; Crossed product C*-algebras |
KK-theory KK-theory is a bivariant homology theory for separable C*-algebras introduced by Gennadi Kasparov that unifies K-theory and K-homology and provides a framework for index theory, duality, and classification problems in operator algebras. It plays a central role in the Baum–Connes conjecture, the study of group C*-algebras associated to discrete groups such as F2, and in connections between topology and noncommutative geometry exemplified by Alain Connes's work. KK-theory interfaces with structures from Atiyah–Singer index theorem, Fredholm operator theory, and the representation theory of locally compact groups.
Introduction
KK-theory arose from efforts by Gennadi Kasparov to extend ideas from Michael Atiyah's topological K-theory and Bott periodicity into the realm of operator algebras and to treat equivariant phenomena related to Kazhdan's property (T), amenability, and the Novikov conjecture. It provides bifunctorial abelian groups KK(A,B) for pairs of separable C*-algebras A and B, built from equivalence classes of Kasparov cycles. The theory has been developed further by contributors such as Brown, Douglas and Fillmore, Nigel Higson, John Roe, and Bruce Blackadar and has been applied to problems in manifold topology, representation theory of Lie groups, and classification of nuclear C*-algebras.
Definitions and basic properties
KK(A,B) is an abelian group functor in the variables A and B: contravariant in A and covariant in B. For separable C*-algebras, KK satisfies homotopy invariance, stability with respect to K(H) on a separable Hilbert space, and Bott periodicity. The theory admits exact triangles analogous to those in triangulated categorys and fits into long exact sequences associated to short exact sequences of C*-algebras. KK-theory recovers ordinary K-theory via KK(C,A) ≅ K0(A) and K-homology via KK(A,C) ≅ K0(A)† for appropriate grading conventions. Important properties include excision for split extensions, functoriality under *-homomorphisms between C*-algebras, and compatibility with crossed products by locally compact groups.
Kasparov's KK-groups and cycles
A Kasparov (A,B)-module or cycle consists of a countably generated graded Hilbert B-module E carrying a graded *-homomorphism φ: A → 𝓛_B(E) together with an operator F in 𝓛_B(E) satisfying relations that generalize Fredholm operator conditions modulo compact operators 𝓚_B(E). Homotopy and operator homotopy generate equivalence classes whose classes form KK(A,B). In the equivariant setting one considers actions of a locally compact group G and obtains G-equivariant groups KK^G(A,B). Kasparov introduced these cycles to treat index-theoretic constructions for families and to encode extension data such as those studied in the Brown–Douglas–Fillmore program.
Functoriality and exact sequences
KK-theory is functorial with respect to *-homomorphisms: a morphism f: A1 → A2 induces maps f^*: KK(A2,B) → KK(A1,B) and f_*: KK(A,B1) → KK(A,B2). These maps are compatible with the Kasparov product and with suspension isomorphisms. For a short exact sequence 0 → I → A → A/I → 0 of separable C*-algebras, KK yields six-term exact sequences linking KK groups for I, A, and A/I, mirroring the six-term exact sequence in K-theory and enabling Mayer–Vietoris type arguments for gluings appearing in the study of groupoid C*-algebras and foliation C*-algebras.
Connections with K-theory and K-homology
KK-theory simultaneously generalizes K-theory and K-homology: the groups K0(A) and K1(A) are realized as KK^0(C,A) and KK^1(C,A) respectively, while analytic K-homology classes of a compact Spin^c manifold M appear in KK(C(M),C). The Kasparov product pairs KK(A,B) with KK(B,C) to recover index pairings such as the analytic index of elliptic operators featured in the Atiyah–Singer index theorem. This framework clarifies the role of assembly maps in the Baum–Connes conjecture and connects with the Rosenberg index obstruction and coarse geometry results of John Roe.
Products and the Kasparov product
The associative Kasparov product KK(A,B) × KK(B,C) → KK(A,C) is the central multiplicative structure of the theory, analogous to composition of correspondences in algebraic geometry and to the product in bivariant K-theory notions. Construction of the product uses interior tensor products of Hilbert modules and operator homotopy techniques; it is compatible with external products, Bott periodicity, and descent maps for crossed products by locally compact groups. The product yields Poincaré duality phenomena for pairs like (C(X),C_0(T^*X)) where X is a compact manifold, and underpins dualities used in classification results for nuclear C*-algebras by Elliott-style invariants.
Applications and examples
KK-theory has applications to the Baum–Connes conjecture for discrete groups such as SL(2,Z), to the classification of nuclear C*-algebras via the Elliott program, and to index theory for covering spaces and foliations modeled by groupoids. Concrete examples include computations of KK for commutative algebras C(X) related to topological K-theory of spaces X, KK-groups for rotation algebras (irrational rotation algebras) and noncommutative tori studied by Marc Rieffel, and KK-theory computations for reduced group C*-algebras of hyperbolic groups investigated by Gromov-inspired approaches. KK provides tools for proving permanence properties like nuclearity and exactness for crossed products by amenable group actions and for constructing counterexamples in classification via Kirchberg–Phillips type theorems.
Advanced topics and generalizations
Generalizations include equivariant KK^G for locally compact groups G, real KK-theory for Real C*-algebras related to KR-theory of Michael Atiyah, unbounded KK-cycles linking to spectral triple frameworks of Alain Connes, and bivariant cyclic theories connecting to Cyclic cohomology and the Chern character. Further extensions consider triangulated category structures, E-theory introduced by Connes and Higson, and categories of correspondences used in noncommutative geometry and homotopy-theoretic formulations connected to stable homotopy theory and motivic analogues. Advanced analytic developments involve assembly maps, descent for crossed products, and applications to the Novikov conjecture for groups like those with finite asymptotic dimension studied by Guoliang Yu.