\(C^*\)-complexes de Fredholm. II (Q1085866)

Some general theorems concerning the new variants of the Novikov conjecture and the positive scalar curvature problem are investigated. Recently the theory of elliptic operators has been developed in some different directions: \textit{A. Connes} [Operator algebras and applications, Proc. Symp. Pure Math. 38, Part 1, Kingston/Ont. 1980, 521- 628 (1982; Zbl 0531.57023)], on the one hand, has developed the index theory for families of elliptic operators to the case of pseudodifferential operators elliptic along the leaves of a foliation. On the other hand, \textit{A. S. Mishchenko} and \textit{A. T. Fomenko} [Izv. Akad. Nauk SSSR, Ser. Mat. 43, 831-859 (1979; Zbl 0416.46052)] have developed a theory of pseudodifferential operators with coefficients in a fixed \(C^*\)-algebra. Starting from these data and the need of harmonic analysis on Lie groups, the author of this article has combined these two extremes to give an index theory for pseudodifferential operators with coefficients in a continuous field of \(C^*\)-algebras trivial along the fibers of a foliation [see part I in Acta Math. Vietnam. 9, 121-129 (1984; Zbl 0615.58037)]. The author now aplies this theory to some problems of geometry: the homotopic invariance of higher signatures (S. P. Novikov's conjecture) and the positive scalar curvature problem, on foliations, from the point of view of continuous fields of \(C^*\)-algebras on foliations. The main results of A. Connes, P. Baum and of J. Rosenberg on these subjects are fulfilled.