Pseudodifferential extensions and adiabatic deformation of smooth groupoid actions

Abstract

The adiabatic groupoid ${\mathcal{G}}_{ad}$ of a smooth groupoid $\mathcal{G}$ is a deformation relating $\mathcal{G}$ with its algebroid. In a previous work, we constructed a natural action of $\mathbb{R}$ on the C*-algebra of zero order pseudodifferential operators on $\mathcal{G}$ and identified the crossed product with a natural ideal $J(\mathcal{G})$ of $C^{\ast }({\mathcal{G}}_{ad})$. In the present paper we show that $C^{\ast }({\mathcal{G}}_{ad})$ itself is a pseudodifferential extension of this crossed product in a sense introduced by Saad Baaj. Let us point out that we prove our results in a slightly more general situation: the smooth groupoid $\mathcal{G}$ is assumed to act on a C*-algebra $A$. We construct in this generalized setting the extension of order $0$ pseudodifferential operators $\Psi (A,\mathcal{G})$ of the associated crossed product $A⋊\mathcal{G}$. We show that $\mathbb{R}$ acts naturally on $\Psi (A,\mathcal{G})$ and identify the crossed product of $A$ by the action of the adiabatic groupoid ${\mathcal{G}}_{ad}$ with an extension of the crossed product $\Psi (A,\mathcal{G})⋊\mathbb{R}$. Note that our construction of $\Psi (A,\mathcal{G})$ unifies the ones of Connes (case $A=\mathbb{C}$) and of Baaj ($\mathcal{G}$ is a Lie group).

appears in Bulletin des sciences mathématiques (2015)