Foliation C⁎C⁎-algebras on multiply fibred manifolds

Motivated by index theory for semisimple groups, we study the relationship between the foliation C⁎C⁎-algebras on manifolds admitting multiple fibrations. Let F1,…,FrF1,…,Fr be a collection of smooth foliations of a manifold XX. We impose a condition of local homogeneity on these foliations which ensures that they generate a foliation FF under Lie bracket of tangential vector fields. We then show that the product of longitudinal smoothing operators Ψc−∞(F1)⋯Ψc−∞(Fr) belongs to the C⁎C⁎-closure of Ψc−∞(F). An application to noncommutative harmonic analysis on compact Lie groups is presented.