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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 orderpseudodifferential operators on $\mathcal{G}$ and identified the crossedproduct with a natural ideal $J(\mathcal{G})$ of $C^*(\mathcal{G}_{ad})$. Inthe present paper we show that $C^*(\mathcal{G}_{ad})$ itself is apseudodifferential extension of this crossed product in a sense introduced bySaad Baaj. Let us point out that we prove our results in a slightly moregeneral situation: the smooth groupoid $\mathcal{G}$ is assumed to act on aC*-algebra $A$. We construct in this generalized setting the extension of order$0$ pseudodifferential operators $\Psi(A,\mathcal{G})$ of the associatedcrossed product $A\rtimes \mathcal{G}$. We show that $\mathbb{R}$ actsnaturally on $\Psi(A,\mathcal{G})$ and identify the crossed product of $A$ bythe action of the adiabatic groupoid $\mathcal{G}_{ad}$ with an extension ofthe crossed product $\Psi(A,\mathcal{G})\rtimes \mathbb{R}$. Note that ourconstruction of $\Psi(A,\mathcal{G})$ unifies the ones of Connes (case$A=\mathbb{C} $) and of Baaj ($\mathcal{G}$ is a Lie group).

Pseudodifferential extensions and adiabatic deformation of smooth groupoid actions