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Equivariant KK-theory and the Novikov conjecture. (English) Zbl 0647.46053
This fundamental paper is the definitive version with complete proofs of a “conspectus” containing a sketch of the proofs circulated since 1981. The author constructs a G-equivariant bivariant K-theory KK G(A,B) where the arguments are C *-algebras with a continuous G-action and G is a locally compact (not necessarily compactö) group. This theory is then applied to the Novikov conjecture. The author proves a result which contains and considerably strengthens all previously known results on this conjecture.
As in ordinary Kasparov theory the main technical tool is the intersection product KK G(A,B)$$\times KK$$ G(B,C)$$\to KK$$ G(A,C) and so called Dirac- and dual Dirac elements where the first one comes from a Dirac operator and the second one is (in good cases) a right inverse.
The author’s approach actually proves stronger versions of the Novikov conjecture (which imply the ordinary conjecture) for a large class of fundamental groups $$\pi$$, for instance the following conjecture $$SNC_{\beta}:$$ The natural homomorphism $$\beta$$ from the representable K-homology of the classifying space $$B\pi$$ to the K-theory of the group-C *-algebra C *($$\pi)$$ is split injective.
Here is then a sample result: Theorem 6.7. Suppose that $$\pi$$ is a discrete group such that the universal covering space $$X=E\pi$$ of the classifying space $$B\pi$$ is a special $$\pi$$-manifold (i.e. admits a “dual Dirac” element). Then $$SNC_{\beta}$$ is true for $$\pi$$. In particular, $$SNC_{\beta}$$ is true for all groups $$\pi$$ for which $$B\pi$$ can be taken as a complete Riemannian manifold of non-positive sectional curvature, and also for all closed, discrete, torsionless subgroups of finite component Lie groups.
Independently of its applications to the Novikov conjecture, the equivariant KK G-theory developed in this paper is a powerful tool and an important achievement by itself. It allows, for instance, to mention only one thing, to define a “topological representation ring” $$R(G)=KK$$ G($${\mathbb{C}},{\mathbb{C}})$$ for a locally compact group G and also is the right setting to study the K-theory of so called crossed product C *-algebras. It is impossible, in a short review, to give an adequate idea of the wealth of ideas contained in this article.
Reviewer: J.Cuntz

##### MSC:
 46L80 $$K$$-theory and operator algebras (including cyclic theory) 46L55 Noncommutative dynamical systems 57R20 Characteristic classes and numbers in differential topology 18F25 Algebraic $$K$$-theory and $$L$$-theory (category-theoretic aspects)
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