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 $C^*(\mathcal{ G}_{ ad})$. In the present paper we show that $C^*(\mathcal{ G}_{ ad})$ 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 $A\rtimes \mathcal{ G}$. 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 $\Psi(A,\mathcal{ G})\rtimes \mathbb{ R}$. 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).
Categories: math.OA