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The longitudinal index theorem for foliations. (English) Zbl 0575.58030
A topological formula for the analytical index of a differential operator D elliptic along the leaves of the foliation (V,F) is given. The index in the Atiyah-Singer theorem for families is an element of the $$K^ 0(B)=K_ 0(C(B))$$, where B is the basic space of the fibration. On the case of foliations the algebra C(B) is replaced by a canonically defined $$C^*$$-algebra $$C^*(V;F)$$ and $$ind(D)\in K_ 0(C^*(V;F))$$. The Kasparov K-bifunctor is a basic tool to prove the results of the paper.
Reviewer: V.Deundjak

##### MSC:
 58J20 Index theory and related fixed-point theorems on manifolds 37C85 Dynamics induced by group actions other than $$\mathbb{Z}$$ and $$\mathbb{R}$$, and $$\mathbb{C}$$
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