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Cyclic cohomology and the transverse fundamental class of a foliation. (English) Zbl 0647.46054
Geometric methods in operator algebras, Proc. US-Jap. Semin., Kyoto/Jap. 1983, Pitman Res. Notes Math. Ser. 123, 52-144 (1986).
[For the entire collection see Zbl 0632.00012.]
Let $$(V,F)$$ be a transversally oriented manifold and $$C^*(V,F)$$ the associated $$C^*$$-algebra. The author constructs a “transverse fundamental class”as a cyclic cocycle and shows that this cocycle (defined on a dense subalgebra of $$C^*(V,F))$$ gives rise to a well defined map: $$K_*(C^*(V,F))\to {\mathbb{C}}.$$
The proof uses an analysis of cyclic cocycles satisfying certain continuity conditions (so-called n-traces) on subalgebras of $$C^*$$- algebras and ideas of R. Zimmer, G. Mackey and G. Kasparov to reduce the problem to the special case, where the foliation admits a holonomy invariant “almost isometric” transverse structure.
The general results have implications for the rational injectivity of the natural map $$\mu: K^*(V,F)\to K_*(C^*(V,F))$$ where $$K^*(V,F)$$ is the geometric K-theory of $$(V,F)$$. Other applications are an obstruction for the existence of a metric on F with strictly positive scalar curvature in terms of $$\hat A(V)$$ (generalizing a result of A. Lichnerowicz) and a generalization of a theorem of S. Hurder on connections between the Godbillon-Vey invariant of $$(V,F)$$ and the type of the von Neumann algebra associated with $$(V,F)$$.
A last application concerns the question whether $$[1_ A]$$ is a torsion element in the $$K_ 0$$-group of the crossed product algebra $${\mathcal C}(M)\rtimes \Gamma$$, where $$\Gamma$$ is a discrete group acting on a compact manifold M.
Reviewer: J.Cuntz

##### MSC:
 46L80 $$K$$-theory and operator algebras (including cyclic theory) 18F25 Algebraic $$K$$-theory and $$L$$-theory (category-theoretic aspects) 57R32 Classifying spaces for foliations; Gelfand-Fuks cohomology