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

TL;DR

This paper investigates the spectral properties of Laplacians linked to smooth distributions on compact manifolds, demonstrating their operator nature within foliation $C^*$-algebras under certain smoothness assumptions.

Contribution

It extends previous work by proving that such Laplacians are unbounded regular self-adjoint operators in Hilbert modules over foliation $C^*$-algebras when the associated foliation is smooth.

Findings

Laplacians define unbounded regular self-adjoint operators

Results apply to distributions generating smooth foliations

Advances understanding of spectral properties in noncommutative geometry

Abstract

In this paper we continue the study of spectral properties of Laplacians associated with an arbitrary smooth distribution on a compact manifold, initiated in a previous paper. Under assumption that the singular foliation generated by the distribution is smooth, we prove that the Laplacian associated with the distribution defines an unbounded regular self-adjoint operator in some Hilbert module over the foliation $C^*$-algebra.