The smooth algebra of a one-dimensional singular foliation

Abstract: Androulidakis and Skandalis showed how to associate a holonomy groupoid, a smooth convolution algebra and a C*-algebra to any singular foliation. In this note, we consider the singular foliations of a one-dimensional manifold given by vector fields that vanish to order k at a point. We show that, whereas the C*-algebras of these foliations are divided into two isomorphism classes according to the parity of k, the smooth algebras are pairwise nonisomorphic. This is accomplished by analyzing certain natural ideals in the smooth algebras. Issues of factorization with respect to convolution arise and are resolved using a context-appropriate version of the Diximier-Malliavin theorem.