Foliation C*-algebras on multiply fibred manifolds

Motivated by index theory for semisimple groups, we study the relationship between the foliation C^-algebras on manifolds admitting multiple fibrations. Let F_1,…,F_r be a collection of smooth foliations of a manifold X. We impose a condition of local homegeneity on these foliations which ensures that they generate a foliation F under Lie bracket of tangential vector fields. We then show that the product of longitudinal smoothing operators along each F_j belongs to the C-closure of the smoothing operators along F. An application to noncommutative harmonic analysis on compact Lie groups is presented.