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 $Ψ(A,\mathcal{G})$ of the associated crossed product $A\rtimes \mathcal{G}$. We show that $\mathbb{R}$ acts naturally on $Ψ(A,\mathcal{G})$ and identify the crossed product of $A$ by the action of the adiabatic groupoid $\mathcal{G}_{ad}$ with an extension of the crossed product $Ψ(A,\mathcal{G})\rtimes \mathbb{R}$. Note that our construction of $Ψ(A,\mathcal{G})$ unifies the ones of Connes (case $A=\mathbb{C} $) and of Baaj ($\mathcal{G}$ is a Lie group).
Gluing action groupoids: Fredholm conditions and layer potentials
本文是arXiv：1807.05418和arXiv：1808.01442的合并。我们引入了一类新的群胚，称为“边界作用群胚”，这是通过粘合约化的作用群胚。我们表明，这样的群胚模型分析许多奇异空间，我们给出了几个例子。在群胚作用的一定条件下，得到了边界作用群胚生成的伪微分算子的Fredholm判别准则。此外，我们证明了在早期的论文中构造的锥形域的层势群胚（Carvalho-Qiao，Central European J. Math.，2013）都是Fredholm群胚和边界作用群胚，这使我们能够以统一的方式处理奇异空间上的许多分析问题。作为应用，我们得到了层势群胚上算子的Fredholm准则.
