2020/09/29 by Michael Francis, Francis, Michael · 1 citation
Mathematics · #Advanced Topics in Algebra #Advanced Topology and Set Theory #Homotopy and Cohomology in Algebraic Topology #math.DG #math.OA
paper · pdf · doi:10.48550/arxiv.2009.13760
24 pages
arxiv created 2020/09/29 · arxiv updated 2020/09/30
A famous theorem of Dixmier-Malliavin asserts that every smooth, compactly-supported function on a Lie group can be expressed as a finite sum in which each term is the convolution, with respect to Haar measure, of two such functions. We establish that the same holds for a Lie groupoid. Most of the heavy lifting is done by a lemma in the original work of Dixmier-Malliavin. We also need the technology of Lie algebroids and the corresponding notion of exponential map. As an application, we obtain a result on the arithmetic of ideals in the smooth convolution algebra of a Lie groupoid arising from functions vanishing to given order on an invariant submanifold of the unit space.