The smooth algebra of a one-dimensional singular foliation

TL;DR

This paper investigates the smooth and C*-algebras associated with one-dimensional singular foliations defined by vector fields vanishing to order k, revealing that smooth algebras are pairwise nonisomorphic while C*-algebras fall into two classes based on parity of k.

Contribution

It demonstrates the nonisomorphism of smooth algebras for these foliations and analyzes their ideal structure, extending the understanding of algebraic invariants in singular foliations.

Findings

Smooth algebras are pairwise nonisomorphic for different k.

C*-algebras are classified into two isomorphism classes based on the parity of k.

Resolution of convolution factorization issues using a version of the Dixmier-Malliavin theorem.

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.