Laplacians on Smooth Distributions as C*-Algebra Multipliers

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.