Journal of Operator Theory

Volume 94, Issue 2, Autumn 2025 pp. 473-511.

Filtered calculus and crossed products by $\mathbb R$-actions

Authors: Clement Cren
Author institution: Mathematisches Institut, Georg-August Universitaet Goettingen, Germany

Summary: We show an isomorphism between the kernel of the $C^*$-algebra of the tangent groupoid of a filtered manifold and the crossed product of the order $0$ pseudodifferential operators in the associated filtered calculus by a natural $\mathbb R$-action. This isomorphism is constructed in the same way as in the classical pseudodifferential calculus by Debord and Skandalis. The proof however relies on a structure result for the $C^*$-algebra of graded nilpotent Lie groups which did not appear in the commutative case and extends a result of Epstein and Melrose in the case of contact manifolds.

DOI: http://dx.doi.org/10.7900/jot.2025feb07.2487
Keywords: pseudodifferential calculus, groupoids, non-commutative geometry, analysis on Lie groups

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