Foliation C*-algebras on multiply fibred manifolds

TL;DR

This paper investigates the structure of foliation C*-algebras on manifolds with multiple fibrations, establishing how products of smoothing operators relate to the generated foliation and applying results to noncommutative harmonic analysis.

Contribution

It introduces a condition of local homogeneity for multiple foliations and shows how their associated smoothing operators relate within the C*-algebra framework, with applications to harmonic analysis.

Findings

Product of longitudinal smoothing operators along each foliation lies in the C*-closure of smoothing operators along the generated foliation.

The local homogeneity condition ensures the generated foliation from multiple fibrations.

Application to noncommutative harmonic analysis on compact Lie groups.

Abstract

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.