In this paper, the authors obtain necessary and sufficient conditions for classes of operators to be Fredholm. Their results specialise to yield Fredholm conditions for classical pseudodifferential operators on manifolds with cylindrical and poly-cylindrical ends, on manifolds that are asymptotically Euclidean, and on manifolds that are asymptotically hyperbolic. Other examples of non-compact manifolds covered by their results include natural operators on spaces obtained by desingularisation of suitable singular spaces by successively blowing up the lowest-dimensional singular strata.
The paper consists of roughly two parts: the theoretical part and applications. The applications are included in the last two sections. The most concrete applications are contained in the last section, which the authors tried to write in such a way that it can, to a large extent, be read independently of the rest of the paper.
The paper is organised as follows. The first section is an introduction to the subject. The authors start with reviewing the relevant topics related to groupoids and groupoid \(\mathcal{C}^*\)-algebras, so Section 2 consists mostly of background material on locally compact groupoids \(\mathcal{G}\), their Haar systems, and their \(\mathcal{C}^*\)-algebras. They then review manifolds with corners and tame submersions, and recall the definitions of Lie groupoids in the framework that they need, that is, that of manifolds with corners. All their Lie groupoids are second countable and Hausdorff. They finish this section with a review of some examples of Lie groupoids that play a role in what follows. Section 3 contains preliminaries on exhaustive families of representations and some general results on groupoid \(\mathcal{C}^*\)-algebras. The authors recall the notions of strictly spectral and strictly norming families and the main results. They then introduce groupoids with Exel’s property and with the strong Exel property, respectively. A groupoid satisfies Exel’s property (respectively, strong Exel’s property) when its family of regular representations is exhaustive for the reduced \(\mathcal{C}^*\)-algebra (respectively, for the full \(\mathcal{C}^*\)-algebra). They prove that groupoids given by fibred pull-backs of bundles of amenable Lie groups always have Exel’s strong property. Motivated by this result, they introduce the class of stratified submersion groupoids, given essentially by gluing fibred pull-backs of bundles of Lie groups. They prove that stratified submersion Lie groupoids with amenable isotropy groups always have Exel’s strong property. In Section 4, the authors define Fredholm Lie groupoids and always work in the setting of Lie groupoids, except where otherwise explicitly stated. They provide a characterisation of Fredholm Lie groupoids using strictly spectral families of regular representations and show that groupoids that have the strong Exel’s property are Fredholm. It follows that stratified submersion Lie groupoids with amenable isotropy groups are Fredholm. They specialise to algebras of pseudo-differential operators on Lie groupoids and obtain the crucial results. The final section of the paper, Section 5, contains examples and applications of their results.
For the entire collection see [Zbl 1401.47001].