Journal of Lie Theory 32 (2022), No. 3, 879--898
Copyright Heldermann Verlag 2022
A Dixmier-Malliavin Theorem for Lie Groupoids
Michael D. Francis
Department of Mathematics, University of Western Ontario, Middlesex College, London, Canada
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. The analytical 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.
Keywords: Dixmier-Malliavin, Lie groupoid, convolution.
MSC: 22A22, 58H05.
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