A Dixmier-Malliavin theorem for Lie groupoids

Michael Francis

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arXiv · 2009.13760

A Dixmier-Malliavin theorem for Lie groupoids

Michael Francis

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Abstract

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.

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Michael Francis. 2020-09-29. A Dixmier-Malliavin theorem for Lie groupoids. https://arxiv.org/abs/2009.13760

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