  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.
