Abstract
Motivated by index theory for semisimple groups, we study the relationship between the foliation C⁎-algebras on manifolds admitting multiple fibrations. Let F1,…,Fr be a collection of smooth foliations of a manifold X. We impose a condition of local homogeneity 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 Ψc−∞(F1)⋯Ψc−∞(Fr) belongs to the C⁎-closure of Ψc−∞(F). An application to noncommutative harmonic analysis on compact Lie groups is presented.
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