\(C^*\)-complexes de Fredholm. I (Q1820489)

[For part II see Acta Math. Vietnam. 9, 193-199 (1984; Zbl 0608.58040)]. If \(V\) is a smooth manifold with foliation \(F\), then there is the Connes- Skandalis index theorem for pseudodifferential operators which are elliptic in the longitudinal direction; the indexes take values in \(K_ 0(C^*(V/F))\), where \(C^*(V/F)\) is the \(C^*\)-algebra of the foliation. The author gives a topological formula for the analytical index \(Ind(D)\in K^*_ A(V/F)=K_*(C^*_ A(V,F))\) of a pseudodifferential operator D A-elliptic along the leaves of the \((V,F)\), where A is the \(C^*\)-algebra field. The bivariant K-theory of Kasparov is a basic tool to prove this index theorem.