# Laplacians on generalized smooth distributions as $C^*$-algebra multipliers

## TL;DR

This paper explores the spectral properties of Laplacians linked to smooth distributions on compact manifolds, showing under certain conditions they act as multipliers on foliation $C^*$-algebras, with a focus on their construction and analysis.

## Contribution

It introduces a novel connection between Laplacians on smooth distributions and $C^*$-algebra multipliers, extending spectral analysis in this geometric context.

## Key findings

- Laplacians define unbounded multipliers on foliation $C^*$-algebras under regularity assumptions
- Construction of a parametrix for the Laplacian
- Survey of generalized smooth distributions and associated structures

## Abstract

In this paper, we discuss spectral properties of Laplacians associated with an arbitrary smooth distribution on a compact manifold. First, we give a survey of results on generalized smooth distributions on manifolds, Riemannian structures and associated Laplacians. Then, under the assumption that the singular foliation generated by the distribution is regular, we prove that the Laplacian associated with the distribution defines an unbounded multiplier on the foliation $C^*$-algebra. To this end, we give the construction of a parametrix.