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