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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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