Laplacians on Smooth Distributions as $C^*$-Algebra Multipliers

In this paper, we continue the study of spectral properties of Laplacians associated with an arbitrary smooth distribution on a compact manifold started in a previous paper. Under the 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 $C^*$-algebra of the foliation.