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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

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