Operator K-theory and its applications: Elliptic operators, group representations, higher signatures, \(C^*\)-extensions. (English) Zbl 0571.46047

Kasparov, G. G.

Proc. Int. Congr. Math. Warszawa 1983, Vol. 2, 987-1000 (1984).

[For the entire collection see Zbl 0553.00001.]
This article is an exposition of the authors generalized K-theory and its applications. The Kasparov groups, denoted KK(A,B), A and B \(C^*\)- algebras, are an amalgamation of the Atiyah-Hirzebruch K-cohomology groups and the Brown-Douglas-Fillmore K-homology groups. They have proved remarkably useful in operator algebra theory and in index theory of elliptic operators. The theory is particularly good for studying situations where a group acts on the \(C^*\)-algebras involved. Using these groups the author has been able to settle many cases of the Novikov conjecture on homotopy invariance of higher signatures. This, and other applications relating the representation theory of Lie groups are reviewed in the present paper. It provides a very reasonable introduction to the theory.

Reviewer: J.Kaminker

MSC:

46M20	Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.)
46L55	Noncommutative dynamical systems
58J22	Exotic index theories on manifolds
22E45	Representations of Lie and linear algebraic groups over real fields: analytic methods

Keywords:

generalized K-theory; Kasparov groups; amalgamation of the Atiyah- Hirzebruch K-cohomology groups and the Brown-Douglas-Fillmore K-homology groups; index theory of elliptic operators; Novikov conjecture on homotopy invariance of higher signatures; representation theory of Lie groups

Citations:

Zbl 0553.00001