Fredholm conditions on non-compact manifolds: theory and examples

TL;DR

This paper establishes explicit Fredholm conditions for pseudodifferential operators on non-compact and singular manifolds, using the concept of Fredholm groupoids and their representations, with applications to various geometric settings.

Contribution

It introduces the concept of Fredholm groupoids and characterizes Fredholm conditions via ellipticity and invertibility of limit operators, extending to complex geometric structures.

Findings

Fredholm operators are characterized by ellipticity and invertibility of limit operators.

The class of stratified submersion groupoids has Exel's property, ensuring exhaustiveness of regular representations.

Fredholm conditions apply to manifolds with singularities, desingularizations, and blow-ups.

Abstract

We give explicit Fredholm conditions for classes of pseudodifferential operators on suitable singular and non-compact spaces. In particular, we include a "user's guide" to Fredholm conditions on particular classes of manifolds including asymptotically hyperbolic manifolds, asymptotically Euclidean (or conic) manifolds, and manifolds with poly-cylindrical ends. The reader interested in applications should be able read right away the results related to those examples, beginning with Section 5. Our general, theoretical results are that an operator adapted to the geometry is Fredholm if, and only if, it is elliptic and all its limit operators, in a sense to be made precise, are invertible. Central to our theoretical results is the concept of a Fredholm groupoid, which is the class of groupoids for which this characterization of the Fredholm condition is valid. We use the notions of exhaustive and strictly spectral families of representations to obtain a general characterization of Fredholm groupoids. In particular, we introduce the class of the so-called groupoids with Exel's property as the groupoids for which the regular representations are exhaustive. We show that the class of "stratified submersion groupoids" has Exel's property, where stratified submersion groupoids are defined by glueing fibered pull-backs of bundles of Lie groups. We prove that a stratified submersion groupoid is Fredholm whenever its isotropy groups are amenable. Many groupoids, and hence many pseudodifferential operators appearing in practice, fit into this framework. This fact is explored to yield Fredholm conditions not only in the above mentioned classes, but also on manifolds that are obtained by desingularization or by blow-up of singular sets.

Equations

$$\begin{gathered} P:H^s(M_0;E)\to H^{s-m}(M_0;F)\text{mbox}isFredholm\iff P\text{mbox}isellipticand \\ {P}_{\alpha }:H^s(M_{\alpha };E)\to H^{s-m}(M_{\alpha };F)\text{mbox}isinvertibleforall\alpha \in I. \end{gathered}$$

$$a\in 1+C_r^{\ast }(\mathcal{G})$$

$$M\ni x\mapsto {\lambda }_x(\phi ):={\int }_{{\mathcal{G}}_x}\phi (g)\mathrm{d}{\lambda }_x(g)\in \mathbb{C}$$

$${\int }_{{\mathcal{G}}_{r(g)}}\phi (hg)\mathrm{d}{\lambda }_{r(g)}(h)={\int }_{{\mathcal{G}}_{d(g)}}\phi (h)\mathrm{d}{\lambda }_{d(g)}(h)$$

$$({\phi }_1\ast {\phi }_2)(g):=\underset{d^{-1}(d(g))}{\int }{\phi }_1(g{h}^{-1}){\phi }_2(h)\mathrm{d}{\lambda }_{d(g)}(h).$$

$${\phi }^{\ast }(g):=\overline{\phi (g^{-1})}$$

$$\|f\|_{I}\,:=\,\max\,\Bigl{\{}\,\sup_{x\in M}\int|\varphi|{\rm d}\lambda_{x},\,\sup_{x\in M}\int|\varphi^{*}|{\rm d}\lambda_{x}\,\Bigr{\}}.$$

$$\Vert \phi \Vert :=\underset{\pi }{\sup }\Vert \pi (\phi )\Vert ,$$

$${\pi }_x(\phi )\psi (g):=\phi \ast \psi (g):={\int }_{{\mathcal{G}}_{d(g)}}\phi (g{h}^{-1})\psi (h)d{\lambda }_{d(g)}(h),\varphi \in {\mathcal{C}}_c(\mathcal{G}).$$

$$\Vert \phi {\Vert }_r:=\underset{x\in M}{\sup }\Vert {\pi }_x(\phi )\Vert .$$

$$0\to C^{\ast }({\mathcal{G}}_U)\to C^{\ast }(\mathcal{G})\stackrel{{\rho }_F}{⟶}{C}^{\ast }({\mathcal{G}}_F)\to 0.$$

$$0\to C_r^{\ast }({\mathcal{G}}_U)\to C_r^{\ast }(\mathcal{G})\stackrel{({\rho }_F{)}_r}{\to }{C}_r^{\ast }({\mathcal{G}}_F)\to 0.$$

$$(d{h}_x{)}^{-1}(T_{h(x)}^{+}M)=T_x^{+}{M}_1.$$

$$f^{\downarrow \downarrow }(\mathcal{H}):=\{(m,g,m^{\prime })\in M\times \mathcal{H}\times M,f(m)=r(g),d(g)=f(m^{\prime })\},$$

$$A(f^{\downarrow \downarrow }(\mathcal{H}))=\{(\xi ,X)|\xi \in A(\mathcal{H}),X\in TM,\varrho (\xi )=f_{\ast }(X)\},$$

$$\bigcup _{\varphi \in \mathcal{F}}\mathrm{supp}(\varphi )=\bigcup _{\varphi \in \mathcal{F}}\mathrm{Prim}(A)\backslash \mathrm{Prim}(\ker (\varphi ))=\mathrm{Prim}(A)\backslash \bigcap _{\varphi \in \mathcal{F}}\mathrm{Prim}(\ker (\varphi )).$$

$$X=M{\times }_L\mathcal{H}:=\{(m,\gamma )\in M\times \mathcal{H}|f(m)=r(\gamma )\}.$$

$$\varnothing :=U_{-1}\subset U_0\subset \dots \subset U_{i-1}\subset U_i\subset \dots \subset U_N:=M,$$

$${\mathcal{H}}_S\simeq f_S^{\downarrow \downarrow }(G_S).$$

$$\mathcal{H}={\bigsqcup }_S{f}_S^{\downarrow \downarrow }(G_S)={\bigsqcup }_{S}S{\times }_{B}S{\times }_B{G}_S,$$

$$C_r^{\ast }(\mathcal{G})/C_r^{\ast }({\mathcal{G}}_U)\simeq C_r^{\ast }({\mathcal{G}}_F),F:=M\setminus U.$$

$$C_r^{\ast }({\mathcal{G}}_F)\simeq C_r^{\ast }(\mathcal{G})/C_r^{\ast }({\mathcal{G}}_U)\simeq C_r^{\ast }(\mathcal{G})/\mathcal{K}(L^2(U)).$$

$${\Psi }^m(\mathcal{G};E,F):=f{M}_N({\Psi }^m(\mathcal{G}))e.$$

$${\mathrm{Diff}}^m(\mathcal{V};E,F)={\Psi }^m(\mathcal{G};E,F)\cap \mathrm{Diff}(M;E,F),$$

$$0\to C^{\ast }(\mathcal{G})\to \overline{\Psi }(\mathcal{G})\stackrel{{\sigma }_0}{-\to }{\mathcal{C}}_0(S^{\ast }A)\to 0.$$

$$P_{1}\,:=\,\left[\begin{array}[]{ccc}0&P&0\\ P^{*}&0&0\\ 0&0&Q_{m}\end{array}\right]\,,$$

$$\Vert P{\Vert }_{m,s}:=\Vert (1+{\Delta }_F{)}^{(s-m)/2}P(1+{\Delta }_E{)}^{-s/2}{\Vert }_{L^2\to L^2}$$

$$L_s^m(\mathcal{G};E,F):=\Vert \cdot {\Vert }_{m,s}\text{mbox}-closureof{\Psi }^m(\mathcal{G};E,F).$$

$$(1+{\Delta }_F{)}^{(s-m)/2}P(1+{\Delta }_E{)}^{-s/2}\in \overline{\Psi }(\mathcal{G};E,F)=:L_0^0(\mathcal{G};E,F)$$

$${\mathcal{W}}^m(\mathcal{G}):={\Psi }^m(\mathcal{G})+{\cap }_s{L}_s^{-\infty }(\mathcal{G}).$$

Full text

V.N. has been partially supported by ANR-14-CE25-0012-01. Qiao was partially supported by NSF of China (11301317, 11571211). Carvalho was partially supported by Fundação para a Ciência e Tecnologia UID/MAT/04721/2013 (Portugal).

Manuscripts available from http://iecl.univ-lorraine.fr/$\tilde{}$Victor.Nistor/

1. 1 Introduction

2. 1.1 Background and main result

3. 1.2 Exhaustive families of representations

4. 1.3 Fredholm Lie groupoids

5. 1.4 Examples and applications

6. 1.5 Contents of the paper

7. 2 Groupoids and their $C^{*}$-algebras

8. 2.1 Locally compact groupoids

9. 2.2 Haar systems and $C^{\ast}$-algebras

10. 2.3 Manifolds with corners and Lie groupoids

11. 2.4 Examples of Lie groupoids

12. 3 Exhaustive families of representations and Exel’s question

13. 3.1 Exhaustive families of representations of $C^{\ast}$-algebras

14. 3.2 Exel’s question and properties

15. 4 Fredholm conditions

16. 4.1 Fredholm groupoids and their characterization

17. 4.2 Pseudodifferential operators

18. 5 Examples

19. 5.1 Examples related to group actions

20. 5.1.1 The action of a group on a space

21. 5.1.2 The $b$-groupoid

22. 5.1.3 Manifolds with poly-cylindrical ends

23. 5.1.4 Asymptotically Euclidean spaces

24. 5.1.5 Asymptotically hyperbolic manifolds

25. 5.2 The edge groupoid

26. 5.3 Desingularization groupoids

27. 5.4 Desingularization and singular spaces

28. 5.4.1 The desingularization of a smooth submanifold

29. 5.4.2 The desingularization of a submanifold with boundary

30. 5.4.3 Desingularization of a stratified subset of dimension one

1. Introduction

We obtain in this paper necessary and sufficient conditions for classes of operators to be Fredholm. Our results specialize to yield Fredholm conditions for 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 our results include operators obtained by desingularization of suitable singular spaces by successively blowing up the lowest dimensional singular strata.

1.1. Background and main result

Let $M_{0}$ be a Riemannian manifold and let $P\colon H^{s}(M_{0};E)\to H^{s-m}(M_{0};F)$ be an order $m$ pseudo-differential operator acting between Sobolev sections of two smooth, Hermitian vector bundles $E,F$. Recall that $P$ is called elliptic if its principal symbol $\sigma_{m}(P)$ is invertible outside the zero section. When $M_{0}$ is compact, a classical, well-known result [18, 93, 94] states that $P$ is Fredholm if, and only if, it is elliptic. This classical Fredholm result has many applications, so a natural question to ask is to what extent it extends to (suitable) non-compact manifolds. The example of constant coefficient differential operators on ${\mathbb{R}^{n}}$ shows that that this classical result might be no longer true as stated if $M_{0}$ is not compact.

On certain classes of manifolds, however, it is possible to reformulate this classical result as follows. Let $M_{0}$ be a non-compact manifold with “amenable ends” (see Subsection 4.2 for the precise definition). Then we can associate to $M_{0}$ the following data:

1. (1)

Smooth manifolds $M_{\alpha}$, $\alpha\in I$, and 2. (2)

Lie groups $G_{\alpha}$ acting on $M_{\alpha}$, $\alpha\in I$,

(for a suitable index set $I$), satisfying the following theorem.

Theorem 1.1**.**

Let $P$ be an order $m$ pseudodifferential operator on $M_{0}$ compatible with the geometry. Then one can associate to $P$ certain $G_{\alpha}$-invariant pseudodifferential operators $P_{\alpha}$ on $M_{\alpha}$ with the following property:

[TABLE]

Many results of this kind were obtained for psedudodifferential operators before. See for instance [24, 52, 58, 56, 91, 92] and the references therein for a very small sample. Similar results for differential operators were also obtained earlier, see [23, 41, 42, 67, 76] and the references therein, again for a very small sample. Other related results appeared in the context of localization principles, often in relation to the $N$-body problem. See [21, 29, 30, 46, 53, 78, 85] and the references therein. Inspired by these works, we shall call the operators $P_{\alpha}$ limit operators (of $P$). We shall refer to results similar to Theorem 1.1 as non-local Fredholm conditions.

The class of “manifolds with amenable ends” is introduced such that, almost by definition, it is close to being the largest class of manifolds for which Theorem 1.1 is valid. The challenge then becomes to provide a large enough class of manifolds with amenable ends, which we do in this paper. The term “amenable” comes from the fact that certain isotropy groups at infinity must be amenable. In many cases, the manifolds $M_{\alpha}$ are obtained from the orbits of certain natural vector fields acting on a compactification $M$ of $M_{0}$ and from their isotropies. This is the case for manifolds with cylindrical and poly-cylindrical ends, for manifolds that are asymptotically Euclidean, and for manifolds that are asymptotically hyperbolic, which are all particular cases of manifolds with amenable ends. See the last section of the paper for explicit statements covering in detail the case of these classes of manifolds. It is interesting to notice that the case of asymptotically hyperbolic manifolds leads to the study of certain operator on solvable Lie groups.

Theorem 1.1 is almost a consequence of the results in [44, 45], but the results of those papers turned out to be difficult to use by non-specialists. We have thus tried in the last section of the paper to provide a presentation such that the reader can understand the main results without any knowledge of groupoids or $C^{\ast}$-algebras.

1.2. Exhaustive families of representations

It was realized by some of the authors mentioned above that the non-local Fredholm conditions of Theorem 1.1 are related to the representation theory of certain $C^{\ast}$-algebras $\mathfrak{A}$. See also [17, 18, 76, 98] and the references therein. These $C^{\ast}$-algebras are such that they contain the operators of the form $a=(1+\Delta)^{(s-m)/2}D(1+\Delta)^{-s/2}$ (where $\Delta$ is the positive Laplacian) acting on $L^{2}(M_{0})$. The reason for considering the operator $a$ is that $D$ is Fredholm if, and only if, $a$ is Fredholm; moreover, $a$ is bounded. In most practical applications, these $C^{\ast}$-algebras can be chosen to be groupoid $C^{\ast}$-algebras. These ideas are used in the proof of Theorem 4.12, which contains Theorem 1.1 as a particular case. The operators $P_{\alpha}$ in Theorem 1.1 then can be obtained as homomorphic images of the operator $P$, that is $P_{\alpha}=\pi_{\alpha}(P)$.

The relevant families of representations in this setting are those of strictly spectral (and strictly norming) families of representations, introduced by Roch in [84] and recently studied in detail by Nistor and Prudhon in [72]. In that paper, the concept of an exhaustive family of representations has emerged as a useful concept to study strictly spectral families of representations. From a technical point of view, these families of representations form the backbone of the theoretical results in this paper.

1.3. Fredholm Lie groupoids

The key point of our approach is to start with a general study of Fredholm conditions for pseudodifferential operators in the framework of Fredholm Lie groupoids. A Fredholm Lie groupoid $\mathcal{G}$ is, by definition, a Lie groupoid for which the Fredholm property of its associated operators is equivalent to the invertibility of the principal symbol and of its fiberwise boundary restrictions. More precisely, if $M$ is the set of units of $\mathcal{G}$ and if $\mathcal{G}$ is the pair groupoid above $M_{0}:=M\smallsetminus{\partial}M$, we have that $\mathcal{G}$ is a Fredholm Lie groupoid if it has the following property:

[TABLE]

The reader will recognize in condition (1) the type of Fredholm conditions that appeared in Theorem 1.1 (and which are typically used in practice). It turns out that if the defining condition (1) is satisfied (for all $a\in 1+C^{\ast}_{r}(\mathcal{G})$) then it will be satisfied for many other operators associated to $\mathcal{G}$, in particular it will be satisfied for $a=P\in\Psi^{m}(\mathcal{G};E,F)$, a pseudodifferential operators on $\mathcal{G}$ acting between the sections of the vector bundles $E,F\to M$ (however, for pseudodifferential operators, one has to add the ellipticity condition).

The extension of the Fredholm conditions in Equation (1) from operators in $1+C^{\ast}_{r}(\mathcal{G})$ to operators in $\Psi^{m}(\mathcal{G};E,F)$ turns out to be almost automatic and is based on the use of strictly spectral and exhaustive families of representations of groupoid $C^{\ast}$-algebras (see Section 3 for definitions and references). Thus, in our paper, we shift the study of the Fredholm conditions from the study of a single operator to the study of suitable algebras containing it. In particular, it will be enough to study the properties and the representations of regularizing pseudodifferential operators. The Fredholm properties of higher order pseudodifferential operators will then follow simply by including the ellipticity condition. Thus the Fredholm conditions established in this paper will be formulated in terms of representations of groupoid $C^{\ast}$-algebras.

We give various characterizations of Fredholm groupoids and provide methods to prove that individual groupoids are Fredholm. We introduce the class of stratified submersion groupoids, which is built out by glueing fibered pull-backs of bundles of Lie groups. We show that a stratified submersion groupoid with amenable stabilizers is Fredholm and hence that it yields a manifold with amenable ends.

1.4. Examples and applications

The main significance of the class of stratified submersion Lie groupoids is that many of the groupoids that appear in practice exhibit this structure naturally. Typically, using the notations as above, we have that $M$ identifies with a compactification of $M_{0}$ to a manifold with corners, where the behavior at $F=M\setminus M_{0}$ models the behavior at infinity (‘at the ends’). Often the corresponding isotropy groups are amenable. A class of manifolds that fits this perspective is the class of Lie manifolds [44, 45], a few examples of which appear in Section 5.

Moreover, stratified submersion Lie groupoids are also tailored to applications to singular spaces obtained by (iterative) desingularization procedures, desingularization being the analog in the category of groupoids of the blow-up construction in the category of manifolds with corners. In fact, it can be seen that desingularization preserves Fredholm groupoids, and that the class of stratified submersion groupoids is closed under desingularization. While we do not pursue this approach here in full generality, we outline the key ideas involved and give the construction of such a groupoid yielding Fredholm conditions for the edge calculus.

Our results can be extended to products of such manifolds, or to manifolds that locally at infinity are products of such manifolds.

1.5. Contents of the paper

The paper consists of roughly two parts: the theoretical part and applications. The applications are included in the last section, which we tried to write in such a way that they can, to a large extent, be read independently of the rest of the paper. The reader interested only in applications can thus start immediately with Section 5. At least the main results of that section should be understandable without any knowledge of groupoids.

We now describe in detail the main contents of the paper. We start with reviewing the relevant topics related to groupoids and groupoid $C^{*}$-algebras, so Section 2 is mostly background material on locally compact groupoids $\mathcal{G}$, on their Haar systems, and their $C^{*}$-algebras. We then review manifolds with corners and tame submersions, and we recall the definitions of Lie groupoids in the framework that we need, that is, that of manifolds with corners. Note that all our Lie groupoids will be second countable and Hausdorff. We finish this section with a review of some examples of Lie groupoids that will play a role in what follows.

Section 3 contains preliminaries on exhaustive families of representations from [72] and some general results on groupoid $C^{\ast}$-algebras. We recall the notions of strictly spectral and strictly norming families [84] and the main results from [72]. We then introduce groupoids with * Exel’s property*, respectively, strong Exel’s property, when the induced family of regular representations is exhaustive for the reduced $C^{*}$-algebra, respectively, for the full $C^{*}$-algebra. We prove that groupoids given by fibered pull-backs of bundles of amenable Lie groups always have Exel’s strong property, and we introduce the class of stratified submersion groupoids, given essentially by glueing fibered pull-backs of bundles of Lie groups. Our main result here is that stratified submersion Lie groupoids with amenable isotropy groups always have Exel’s strong property.

In Section 4, we define Fredholm Lie groupoids. Note that in the following sections, we will work always in the setting of Lie groupoids. We provide a characterization 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. We then specialize to algebras of pseudodifferential operators on Lie groupoids and obtain the crucial Theorems 4.12 and 4.17.

The last section of the paper, Section 5, contains examples and applications of our results, namely of Theorem 4.17. We start with group actions, define the associated transformation groupoid, and use this construction to describe Fredholm conditions related to several pseudodifferential calculi: the $b$-groupoid that models operators on manifolds with cylindrical and poly-cylindrical ends; the scattering groupoid that models operators on manifolds that are asymptotically Euclidean, and also operators on asymptotically hyperbolic spaces. We then consider the edge calculus and construct a suitable Fredholm stratified submersion groupoid that will recover Fredholm conditions. This is a particular case of a desingularization groupoid, whose construction is outlined in the following subsection. We then give more explicit examples of the desingularization and blow-up process. A first example deals with the blow-up of a smooth submanifold along another smooth, compact manifold. The second example extends this construction to manifolds with boundary. Finally, the last example deals with the iterated blow-up of a singular stratified subset of dimension one. See the main body of the paper for specific references to the existing literature.

For simplicity, in view of the applications considered in this paper, we shall work almost exclusively with Lie groupoids (and their reductions), although some definitions and results are valid in greater generality. For the most part, our manifolds will be Hausdorff and second countable.

2. Groupoids and their $C^{*}$-algebras

We recall in this section some basic definitions and properties of groupoids. Although our interest in applications lies mainly in Lie groupoids, we have found it convenient to consider also the general case of locally compact groupoids, so we shall discuss these two cases in parallel. We refer to Mackenzie’s books [49, 50] for more details and, in general, for a nice introduction to the subject of Lie groups and Lie groupoids, as well as to further references and historical comments. See also [12, 60, 81, 106] for the more specialized issues relating to analytic applications. Most of the needed results can be found also in [69], whose approach we also use here.

2.1. Locally compact groupoids

Let us introduce groupoids as in [12, 50, 81].

Definition 2.1**.**

A groupoid is a pair $\mathcal{G}=(\mathcal{G}^{(1)},\mathcal{G}^{(0)})$, where $\mathcal{G}^{(i)}$, $i=0,1$, are sets, together with structural morphisms

1. (1)

$d,r:\mathcal{G}^{(1)}\to\mathcal{G}^{(0)}$ (the “domain” and “range”), 2. (2)

$\iota:\mathcal{G}^{(1)}\to\mathcal{G}^{(1)}$ (the “inverse”), 3. (3)

$u:\mathcal{G}^{(0)}\to\mathcal{G}^{(1)}$ (the inclusion of units), and 4. (4)

$\mu:\mathcal{G}^{(2)}:=\{(g,h)\in\mathcal{G}^{(1)}\times\mathcal{G}^{(1)}|\,d(g)=r(h)\}\to\mathcal{G}^{(1)}$ (the product),

with the following properties (we write $gh:=\mu(g,h)$, for simplicity):

1. (1)

$d(gh)=d(h)$, $r(gh)=r(g)$ if $d(g)=r(h)$; 2. (2)

$g_{1}(g_{2}g_{3})=(g_{1}g_{2})g_{3}$ for all $g_{i}\in\mathcal{G}$ such that $d(g_{i})=r(g_{i+1})$; 3. (3)

$d(u(x))=x=r(u(x))$, for all $x\in M$. 4. (4)

$gu(d(g))=g$ and $u(r(g))g=g$ for all $g\in\mathcal{G}$. 5. (5)

$g\iota(g)=u(r(g))$ and $\iota(g)g=u(d(g))$ for all $g\in\mathcal{G}$.

Moreover, we see that $\mathcal{G}^{(0)}$ can be regarded as the set of objects of a category with morphisms $\mathcal{G}^{(1)}$. The objects of $\mathcal{G}$ will also be called units and the morphisms of $\mathcal{G}$ will also be called arrows. Actually, a groupoid $\mathcal{G}$ is simply a small category in which every morphism is invertible. (A small category is one whose objects form a set.) For convenience, we shall identify $\mathcal{G}=\mathcal{G}^{(1)}$ and denote $M:=\mathcal{G}^{(0)}$. We shall write $\mathcal{G}\rightrightarrows M$ for a groupoid $\mathcal{G}$ with units $M$. In this paper, $\mathcal{G}$ always will denote a groupoid.

Definition 2.2**.**

A locally compact groupoid is a groupoid $\mathcal{G}\rightrightarrows M$ such that:

1. (1)

$\mathcal{G}$ and $M$ are locally compact spaces, with $M$ Hausdorff; 2. (2)

the structural morphisms $d,r,\iota,u,$ and $\mu$ are continuous; 3. (3)

$d$ is surjective and open.

See Subsection 2.4 for examples of groupoids. Let us notice that the definition requires only the space of units $M$ to be Hausdorff. In this paper, however, all spaces will be assumed or proved to be Hausdorff.

In what follows, we denote by $\mathcal{G}_{A}:=d^{-1}(A)$, $\mathcal{G}^{A}=r^{-1}(A)$ and $\mathcal{G}_{A}^{B}:=d^{-1}(A)\cap r^{-1}(B)$. We call $\mathcal{G}_{A}^{A}$ the reduction of $\mathcal{G}$ to $A$. If $A$ is $\mathcal{G}$-invariant, in the sense that $\mathcal{G}_{A}^{A}=\mathcal{G}_{A}=\mathcal{G}^{A}=r^{-1}(A)$, then $\mathcal{G}_{A}$ is also a groupoid, called the restriction of $\mathcal{G}$ to $A$. For any $x\in X$, $\mathcal{G}_{x}^{x}=d^{-1}(x)\cap r^{-1}(x)$ is a group, called the isotropy group at $x$.

We will see several examples of groupoids in more detail when we introduce Lie groupoids.

2.2. Haar systems and $C^{\ast}$-algebras

We now recall the definition of a Haar system of a locally compact groupoid and we use this opportunity to fix some more notations to be used throughout the rest of the paper. We refer to [12, 39, 81] for more information on the topics discussed in this subsection.

If $\mathcal{G}$ is a locally compact groupoid, we shall denote by $\mathcal{C}_{c}(\mathcal{G})$ the space of continuous, complex valued, compactly supported functions on $\mathcal{G}$.

Definition 2.3**.**

A right Haar system for a locally compact groupoid $\mathcal{G}$ is a family $\lambda=\{\lambda_{x}\}_{x\in M}$, where $\lambda_{x}$ is a Borel regular measure on $\mathcal{G}$ with support $\operatorname{supp}(\lambda_{x})=d^{-1}(x)=:\mathcal{G}_{x}$ for every $x\in M=\mathcal{G}^{(0)}$, satisfying

1. (i)

The continuity condition:

[TABLE]

is continuous for $\varphi\in\mathcal{C}_{c}(\mathcal{G})$. 2. (ii)

The invariance condition:

[TABLE]

for all $g\in\mathcal{G}$ and $\varphi\in\mathcal{C}_{c}(\mathcal{G})$.

We shall assume from now on that all our locally compact groupoids are endowed with a (right) Haar system. Let thus $\mathcal{G}$ be a locally compact groupoid and $\{\lambda_{x}\}$ be the Haar system associated to it. The space $\mathcal{C}_{c}(\mathcal{G})$ has a natural product given by the formula

[TABLE]

This makes $\mathcal{C}_{c}(\mathcal{G})$ into an associative $*$-algebra with the involution defined by

[TABLE]

for all $g\in\mathcal{G}$ and $\varphi\in\mathcal{C}_{c}(\mathcal{G})$. There also exists a natural algebra norm on $\mathcal{C}_{c}(\mathcal{G})$ defined by

[TABLE]

The completion of $\mathcal{C}_{c}(\mathcal{G})$ with respect to this norm $\|\cdot\|_{I}$ will be denoted by $L^{1}(\mathcal{G})$.

Recall [26] that a $C^{\ast}$-algebra is a complex algebra $A$ together with a conjugate linear involution $*$ and a complete norm $\|\ \|$ such that $(ab)^{*}=b^{*}a^{*}$, $\|ab\|\leq\|a\|\|b\|$, and $\|a^{*}a\|=\|a\|^{2}$, for all $a,b\in A$. Let $\mathcal{H}$ be a Hilbert space and denote by $\mathcal{L}(\mathcal{H})$ the space of linear, bounded operators on $\mathcal{H}$. Then $\mathcal{L}(\mathcal{H})$ is a $C^{*}$-algebra. A representation of a $C^{\ast}$-algebra $A$ on the Hilbert space $\mathcal{H}_{\pi}$ is a $*$-morphism $\pi:A\to\mathcal{L}(\mathcal{H}_{\pi})$.

To a locally compact groupoid $\mathcal{G}$ (endowed with a Haar system), there are associated two basic $C^{*}$-algebras, the full and reduced $C^{*}$-algebras $C^{\ast}(\mathcal{G})$ and $C_{r}^{\ast}(\mathcal{G})$, whose definition we now recall.

Definition 2.4**.**

The (full) $C^{\ast}$-algebra associated to $\mathcal{G}$, denoted $C^{\ast}(\mathcal{G})$, is defined as the completion of $\mathcal{C}_{c}(\mathcal{G})$ with respect to the norm

[TABLE]

where $\pi$ ranges over all contractive $*$-representations of $\mathcal{C}_{c}(\mathcal{G})$. Let us define as usual for any $x\in M$ the regular representation $\pi_{x}\,\colon\,C^{\ast}(\mathcal{G})\to\mathcal{L}(L^{2}(\mathcal{G}_{x},\lambda_{x}))$ by the formula

[TABLE]

We then define similarly the reduced $C^{\ast}$-algebra $C^{\ast}_{r}(\mathcal{G})$ as the completion of $\mathcal{C}_{c}(\mathcal{G})$ with respect to the norm

[TABLE]

The groupoid $\mathcal{G}$ is said to be metrically amenable if the canonical surjective $*$-homomorphism $C^{\ast}(\mathcal{G})\to C^{\ast}_{r}(\mathcal{G})$, induced by the definitions above, is also injective.

Also, for further use, we note that if $\mathcal{G}$ is second countable, then $C^{\ast}(\mathcal{G})$ is a separable $C^{\ast}$-algebra.

Remark 2.5*.*

For any $\mathcal{G}$-invariant, locally closed subset $A\subseteq M$, the reduced groupoid $\mathcal{G}_{A}=\mathcal{G}_{A}^{A}$ is locally compact and has a Haar system $\lambda_{A}$ obtained by restricting the Haar system $\lambda$ of $\mathcal{G}$ to $\mathcal{G}_{A}$. In particular, we can construct as above the corresponding $C^{\ast}$-algebra $C^{\ast}(\mathcal{G}_{A})$ and the reduced $C^{\ast}$-algebra $C^{\ast}_{r}(\mathcal{G}_{A})$. For any closed subset $A\subseteq M$, the subset $d^{-1}(A)\subseteq\mathcal{G}$ is also closed, so the restriction map $\mathcal{C}_{c}(\mathcal{G})\to\mathcal{C}_{c}(d^{-1}(A))$ is well defined. If $A$ is also $\mathcal{G}$-invariant then the restriction extends by continuity to both a $*$-homomorphism $\rho_{A}\colon C^{\ast}(\mathcal{G})\to C^{\ast}(\mathcal{G}_{A})$ and a $*$-homomorphism $(\rho_{A})_{r}\colon C^{\ast}_{r}(\mathcal{G})\to C^{\ast}_{r}(\mathcal{G}_{A})$.

We have the following well known, but important result [64, 65, 82] that we record for further reference.

Proposition 2.6**.**

Let $\mathcal{G}\rightrightarrows M$ be a second countable, locally compact groupoid with a Haar system. Let $U\subset M$ be an open $\mathcal{G}$-invariant subset, $F:=M\smallsetminus U$.

1. (i)

$C^{\ast}(\mathcal{G}_{U})$* is a closed two-sided ideal of $C^{\ast}(\mathcal{G})$ that yields the short exact sequence*

[TABLE] 2. (ii)

If $\mathcal{G}_{F}$ is metrically amenable, then one has the exact sequence

[TABLE] 3. (iii)

If the groupoids $\mathcal{G}_{F}$ and $\mathcal{G}_{U}$ (respectively, $\mathcal{G}$) are metrically amenable, then $\mathcal{G}$ (respectively, $\mathcal{G}_{U}$) is also metrically amenable.

Proof.

The first assertion is well known, see for instance [65, Lemma 2.10]. The second statement is in [82, Remark 4.10]. The last part follows from (ii) and the Five Lemma. ∎

Remark 2.7*.*

We notice that the exact sequence of Proposition 2.6 corresponds to a disjoint union decomposition $\mathcal{G}=\mathcal{G}_{F}\sqcup\mathcal{G}_{U}.$

2.3. Manifolds with corners and Lie groupoids

Even if one is interested only in analysis on smooth manifolds (that is, in manifolds without corners or boundary), one important class of applications will be to spaces obtained as the blow-up with respect to suitable submanifolds, which leads to manifolds with corners, as each blow-up increases the highest codimension of corners by one. To model the analysis on these spaces, we need Lie groupoids. The advantage of using Lie groupoids is that they have a distinguished class of Haar systems defined using the smooth structure.

We begin with some background material following [69]. It will be important to distinguish between smooth manifolds without corners (or boundaries) and (smooth) manifolds with corners. The former will be smooth manifolds, while the later will be simply manifolds. Thus, in this paper, a manifold $M$ is a second countable, Hausdorff topological space locally modeled by open subsets of $[-1,1]^{n}$ with smooth coordinate changes. In particular, our manifolds may have corners. By contrast, a smooth manifold will not have corners (or boundary). A point $p\in M$ of a manifold (with corners) is called of depth $k$ if it has a neighborhood $V_{p}$ diffeomorphic to $[0,a)^{k}\times(-a,a)^{n-k}$, $a>0$, by a diffeomorphism $\phi_{p}:V_{p}\to[0,a)^{k}\times(-a,a)^{n-k}$ with $\phi_{p}(p)=0$.

The set of inward pointing tangent vectors in $v\in T_{x}(M)$ define a closed cone denoted $T_{x}^{+}(M)$. A function $f:M\to M_{1}$ between two manifolds with corners will be called smooth if its components are smooth in all coordinate charts. A little bit of extra care is needed here in defining the derivatives. This is illustrated clearly in one dimension: if $f:[0,1]\to\mathbb{R}$, then $f^{\prime}(0):=\lim_{h\to 0,t>0}h^{-1}(f(h)-f(0))$, whereas $f^{\prime}(1):=\lim_{h\to 0,t>0}h^{-1}(f(1)-f(1-h))$. Since $\lim_{h\to 0}h^{-1}(f(x+h)-f(x))=\lim_{h\to 0}h^{-1}(f(x)-f(x-h))=:f^{\prime}(x)$, for $0<x<1$, this defines unambiguously $f^{\prime}(x)$. A similar comment is in order in higher dimensions as well. In local coordinates, a smooth function $[0,1]^{n}\to[0,1]^{k}$ is one that is the restriction of a smooth function $\mathbb{R}^{n}\to\mathbb{R}^{k}$.

Let $M$ and $M_{1}$ be manifolds with corners and $f:M_{1}\to M$ be a smooth map. Then $f$ induces a vector bundle map $df:TM_{1}\to TM$ such that $df(T_{z}^{+}(M_{1}))\subset T_{f(z)}^{+}M$. If the smooth map $f:M_{1}\to M$ is injective, has injective differential $df$, and has locally closed range, then we say that $f(M_{1})$ is a submanifold of $M$. We stress the condition that submanifolds be locally closed (that is, the intersection of an open and a closed subset). Other than this conditions, our concept of submanifold is the most general possible.

Definition 2.8**.**

A tame submersion $h$ between two manifolds with corners $M_{1}$ and $M$ is a smooth map $h:M_{1}\to M$ such that its differential $dh$ is surjective everywhere and

[TABLE]

(That is, $dh(v)$ is an inward pointing vector of $M$ if, and only if, $v$ is an inward pointing vector of $M_{1}$.)

Clearly, if $h:M_{1}\to M$ is a tame submersion of manifolds with corners, then $x$ and $h(x)$ will have the same depth. We have the following well known lemma (see for instance [69]).

Lemma 2.9**.**

Let $h:M_{1}\to M$ be a tame submersion of manifolds with corners.

1. (i)

For $m_{1}\in M_{1}$, there exists an open neighborhood $U$ of $m_{1}$ in $M_{1}$ such that $h(U)$ is open and the restriction of $h$ to $U$ is a ${\mathcal{C}}^{\infty}$ fibration with basis $h(U)$. 2. (ii)

Let $L\subset M$ be a submanifold, then $L_{1}:=h^{-1}(L)$ is a submanifold of $M_{1}$ of rank $\leq$ the rank of $L$.

We now define Lie groupoids roughly by replacing our spaces with manifolds with corners and the continuity conditions with smoothness conditions. The reason why Lie groupoids play an important role in applications is that they model the distribution kernels of many interesting classes of pseudodifferential operators.

Definition 2.10**.**

A Lie groupoid is a groupoid $\mathcal{G}\rightrightarrows M$ such that

1. (i)

$\mathcal{G}$ and $M$ are manifolds with corners, 2. (ii)

the structural morphisms $d,r,\iota,$ and $u$ are smooth, 3. (iii)

$d$ is a tame submersion of manifolds with corners, and 4. (iv)

$\mu$ is smooth.

In particular, the Lie groupoids used in this paper are locally compact groupoids and are second countable and Hausdorff. In different contexts in other papers, it is sometimes useful not to assume the set of arrows $\mathcal{G}$ of a Lie groupoid to be Hausdorff, although the space of units $M$ of a Lie groupoid $\mathcal{G}$ is always assumed to be Hausdorff. Lie groupoids were introduced by Ehresmann. A more general class also useful in applications is that of continuous family groupoids, where we assume continuity only along the units, keeping smoothness along the fibers. See [44, 75].

For the rest of this section, $\mathcal{G}$ will always be a Lie groupoid. In fact, most of the groupoids that we will consider in what follows will be Lie groupoids. However, one has to be careful since the restriction of a Lie groupoid is not necessarily a Lie groupoid itself. Moreover, we shall assume that all our Lie groupoids are Hausdorff.

Since $d$ and $r$ are tame submersions, it follows from Lemma 2.9 that the fibers $\mathcal{G}_{x}:=d^{-1}(x)$, $x\in M$, are smooth manifolds (that is, they have no corners). Similarly, the same lemma implies that the set $\mathcal{G}^{(2)}\subset\mathcal{G}\times\mathcal{G}$ of composable units is a manifold as well (but it may have corners). In particular, it makes sense to consider smooth maps $\mathcal{G}^{(2)}\to\mathcal{G}^{(1)}$.

Remark 2.11*.*

Let $\mathcal{G}\rightrightarrows M$ be a Lie groupoid and let $A(\mathcal{G}):=\cup_{x\in M}T_{x}\mathcal{G}_{x}$, where $\mathcal{G}_{x}:=d^{-1}(x)$, as usual. Let us denote by $f_{*}:TM_{1}\to TM_{2}$ the differential of a smooth map $f:M_{1}\to M_{2}$. Then $A(\mathcal{G})$ is the restriction to units of the vector bundle $\ker(d_{*})$, and hence it has a natural structure of vector bundle over $M$. It is called the Lie algebroid of $\mathcal{G}$. The differential $r_{*}$ of the range map $r:\mathcal{G}\to M$ then induces a map $\varrho:=r_{*}|_{A(\mathcal{G})}\to TM$, called the anchor map of $A(\mathcal{G})$. We let $\operatorname{Lie}(\mathcal{G}):=\varrho(\Gamma(M;A(\mathcal{G})))$ denote the image by $\varrho$ of the space of sections of $A(\mathcal{G})$. It is coincides with the image by $r_{*}$ of the space $d$-vertical (i.e. tangent to the fibers $\mathcal{G}_{x}$ of $d$) vector fields on $\mathcal{G}$ that are invariant for the action of $\mathcal{G}$ on itself by right multiplication. Therefore both $\Gamma(M;A(\mathcal{G}))$ and $\operatorname{Lie}(\mathcal{G}):=\varrho(\Gamma(M;A(\mathcal{G})))$ are Lie algebras for the Lie bracket. Both Lie algebras will play an important role in the analysis of differential operators.

Remark 2.12*.*

We can use the vector bundle $A(\mathcal{G})$ to define a Haar system. Let $D:=|\Lambda^{n}A(\mathcal{G})|$, where $n$ is the dimension of the Lie algebroid of $\mathcal{G}$. The pull-back vector bundle $r^{*}(D)$ is the bundle of 1-densities along the fibers of $d$. A trivialization of $D$ will hence give rise to a right invariant set of measures on $\mathcal{G}_{x}$ and hence to a right Haar system.

We conclude that a Lie groupoid always has a (right) Haar system, and we shall always assume that it is obtained as in the remark above.

2.4. Examples of Lie groupoids

We continue with various examples of constructions of Lie groupoids that will be needed in what follows. Recall that we only consider Hausdorff groupoids in this paper. Also, recall that a Lie groupoid $\mathcal{G}$ is metrically amenable if the canonical morphism $C^{\star}(\mathcal{G})\to C^{\star}_{r}(\mathcal{G})$ is an isomorphism.

Example 2.13*.*

Any Lie group $G$ gives rise to a Lie groupoid with set of units reduced to one point: the identity element of $G$. Thus $d,r:G\to M:=\{e\}$ are constant, $\mu:G\times G\to G$ is the group multiplication, $\iota(g):=g^{-1}$ is the (usual) inverse, and $u:M\to G$ is the inclusion of the unit. We have that the Lie algebroid $A(G)$ of $G$ coincides with the Lie algebra of $G$ (the definition of a Lie algebroid was recalled in Remark 2.11). Hence $\operatorname{Lie}(G):=\Gamma(A(\mathcal{G}))$ is also the Lie algebra of $G$. We have that $G$ is metrically amenable if, and onl if, it is amenable in the usual sense of groups. In particular, if $G$ is solvable, then $G$ is amenable.

Example 2.14*.*

Let $M$ be a manifold with corners (hence Hausdorff by our conventions), let $\mathcal{G}=M$, and $d=r=u=\iota=id_{M}$. Then $\mathcal{G}$ is a Lie groupoid with only units. We shall call a Lie groupoid with these properties a manifold. We have that $A(M)=M\times\{0\}$, that is, the zero vector bundle. We have $\operatorname{Lie}(\mathcal{G})=0$.

Example 2.15*.*

Let $\mathcal{G}_{i}\to M_{i}$, $i=1,2$, be two Lie groupoids. Then $\mathcal{G}_{1}\times\mathcal{G}_{2}$ is a Lie groupoid with units $M_{1}\times M_{2}$ and $A(\mathcal{G}_{1}\times\mathcal{G}_{2})=A(\mathcal{G}_{1})\times A(\mathcal{G}_{2})$. We have that $\operatorname{Lie}(\mathcal{G}_{1}\times\mathcal{G}_{2})$ is a suitable topological tensor product of $\operatorname{Lie}(\mathcal{G}_{1})$ and $\operatorname{Lie}(\mathcal{G}_{2})$.

The product of a manifold with a Lie group is thus again a Lie groupoid. We can “twist” this example to obtain a (smooth) “bundle of Lie groups.”

Example 2.16*.*

Let $G$ be a Lie group with automorphism group $\operatorname{Aut}(G)$ and let $P\to M$ be a smooth, locally trivial, principal $\operatorname{Aut}(G)$-bundle, with $M$ and $P$ manifolds with corners (hence Hausdorff, by our conventions). Then the associated fiber bundle $\mathcal{G}:=P\times_{\operatorname{Aut}(G)}G$ with fiber $G$ is a Lie groupoid with units $M$. A Lie groupoid of this form will be called a smooth bundle of Lie groups. It satisfies $d=r$. If $P\to M$ is also differentiable (with $P$ and $M$ manifolds, possibly with corners), then $\mathcal{G}$ is a Lie groupoid. We have $A(\mathcal{G})\simeq P\times_{\operatorname{Aut}(G)}Lie(G)$ is a smooth bundle of Lie algebras and $\varrho:A(\mathcal{G})\to TM$ is the zero map. The Lie algebra $\operatorname{Lie}(\mathcal{G})$ identifies thus with the family of sections of the bundle of Lie algebras $P\times_{\operatorname{Aut}(G)}Lie(G)$. If $G$ is amenable, then $\mathcal{G}$ is metrically amenable. The fact that $\mathcal{G}$ is metrically amenable when $G$ is amenable will be of crucial importance to us and will be used when the group $G$ is a solvable Lie group.

The following simple example of the “pair groupoid” will be fundamental in what follows.

Example 2.17*.*

Let $M$ be a smooth manifold. (Thus $M$ has no corners, according to our terminology). Then the pair groupoid of $M$ is $\mathcal{G}:=M\times M$. It is a groupoid with units $M$. The structural morphisms are as follows: $d$ is the second projection, $r$ is the first projection, and the product $\mu$ is given by $(m_{1},m_{2})(m_{2},m_{3})=(m_{1},m_{3})$. This determines $u(m)=(m,m)$ and $\iota(m,m^{\prime})=(m^{\prime},m)$. We thus have that the product on ${\mathcal{C}}^{\infty}_{c}(\mathcal{G})={\mathcal{C}}^{\infty}_{c}(M\times M)$ is the product of operators with integral kernels. We have $A(M\times M)=TM$, with anchor map the identity map. One crucial feature of the pair groupoid is that, for any $x$, the regular representation $\pi_{x}$ defines an isomorphism between $C^{\ast}(M\times M)$ and the ideal of compact operators in $\mathcal{L}(L^{2}(M))$. In particular, all pair groupoids are metrically amenable.

Example 2.18*.*

Let $M$ be a smooth manifold (so without corners) and $\tilde{M}$ be it universal covering. Let $\pi_{1}(M)$ be the fundamental group of $M$ associated to some fixed point of $M$. A related example to the pair groupoid is that of the path groupoid $\mathcal{P}(M):=(\tilde{M}\times\tilde{M})/\pi_{1}(M)$ of $M$, which will have the same Lie algebroid as the pair groupoid: $A(\mathcal{P}(M))=TM$. The analysis associated to $\mathcal{P}(M)$ is that of $\pi_{1}(M)$-invariant operators on the covering space $\tilde{M}$, thus quite different to that of the pair groupoid. This underscores the importance of choosing the right integrating groupoid when interested in Analysis.

We extend the example of the pair groupoid by defining fibered pull-back groupoids [36, 37] (we use the terminology in [69], however).

Example 2.19*.*

Let again $M$ and $L$ be manifolds and $f:M\to L$ be continuous map. Let $\mathcal{H}\rightrightarrows L$ be a Lie groupoid. The fibered pull-back groupoid is then

[TABLE]

with units $M$ and product $(m,g,m^{\prime})(m^{\prime},g^{\prime},m^{\prime\prime})=(m,gg^{\prime},m^{\prime\prime})$. We shall also sometimes write $M\times_{f}\mathcal{H}\times_{f}M=f^{\downarrow\downarrow}(\mathcal{H})$ for the fibered pull-back groupoid. Assume now that $f$ is a tame submersion (in particular, that it is smooth). Then $f^{\downarrow\downarrow}(\mathcal{H})$ is a Lie groupoid and its Lie algebroid is

[TABLE]

the thick pull-back Lie algebroid $f^{\downarrow\downarrow}A(\mathcal{H})$ of $A(\mathcal{H})$, see [49, 50].

The following particular example of a fibered pull-back will be useful for the study of the $b$-groupoid $\mathcal{G}_{b}$ in Section 5.

Example 2.20*.*

Let us assume that $M$ is a smooth manifold and let $B$ be its set of connected components. Let $G$ be a Lie group. We let $\mathcal{H}:=B\times G$, the product of a manifold and a Lie group, and $f:M\to B$ be the map that associates to a point its connected component. Then $f^{\downarrow\downarrow}(\mathcal{H})$ is the topological disjoint union of the groupoids $(F\times F)\times G$ (product of the pair groupoid and a Lie group) for $F$ ranging through the connected components of $M$.

3. Exhaustive families of representations and Exel’s question

We next recall some basic facts on exhaustive families of representations following [72]. A groupoid has Exel’s property when its set of regular representations is exhaustive. Exel [27] has asked which groupoids have this property. Establishing that a groupoid has Exel’s property will turn out to be important in establishing Fredholm conditions in the next section. At the end of the section, we prove that groupoids given by fibered pull-backs of bundles of Lie groups always have Exel’s property, which leads us to introduce, and extend this property to, the more general class of stratified submersion groupoids.

3.1. Exhaustive families of representations

of $C^{\ast}$-algebras

A two-sided ideal $I\subset A$ of a $C^{*}$-algebra $A$ is called primitive if it is the kernel of an irreducible representation of $A$. We shall denote by $\operatorname{Prim}(A)$ the set of primitive ideals of $A$. The trivial ideal $A$ (of $A$) is not considered a primitive ideal (of $A$), so $A\notin\operatorname{Prim}(A)$. For any representation $\phi$ of $A$, we define its support $\operatorname{supp}(\phi)\subset\operatorname{Prim}(A)$ as the set of primitive ideals of $A$ containing $\ker(\phi)$. Let $J$ be a closed, two-sided ideal of $A$. Then $\operatorname{Prim}(J)=\{I\in\operatorname{Prim}(A)|\,J\not\subset I\}$ and $\operatorname{Prim}(A/J)$ identifies with $\{I\in\operatorname{Prim}(A)|\,J\subset I\}=\operatorname{Prim}(J)^{c}$. We endow $\operatorname{Prim}(A)$ with the hull-kernel topology, which is the topology whose open sets are those of the form $\operatorname{Prim}(J)$, with $J$ a two-sided ideal of $A$.

We shall need the following definition [72].

Definition 3.1**.**

Let $\mathcal{F}$ be a set of representations of a $C^{\ast}$-algebra $A$. We say that $\mathcal{F}$ is exhaustive if $\operatorname{Prim}(A)=\bigcup_{\phi\in\mathcal{F}}\,\operatorname{supp}(\phi)$.

In practice, we rather need families with the following definition [84].

Definition 3.2**.**

Let $\mathcal{F}$ be a set of representations of a unital $C^{\ast}$-algebra $A$.

1. (i)

We say that $\mathcal{F}$ is strictly norming if, for any $a\in A$, there exists $\phi\in\mathcal{F}$ such that $\|\phi(a)\|=\|a\|$. 2. (ii)

We say that $\mathcal{F}$ is strictly spectral if, for any $a\in A$, we have that $a$ is invertible in $A$ if, and only if, $\phi(a)$ is invertible for all $\phi\in\mathcal{F}$.

The two definitions above can be formulated for sets of morphisms or sets of primitive ideals. We also have that $\mathcal{F}_{*}:=\{\pi\in\mathcal{F}|\,\pi\neq 0\}$ has the same properties (“exhaustive,” “strictly norming,” … ) as $\mathcal{F}$.

If $A$ is non-unital, we modify the last definition as follows [72]. Let $A^{+}:=A\oplus\mathbb{C}$ and $\chi_{0}\colon A^{+}\to\mathbb{C}$ be the morphism defined by $\chi_{0}=0$ on $A$ and $\chi_{0}(1)=1$. We then replace $A$ with $A^{+}$ and $\mathcal{F}$ with $\mathcal{F}^{+}:=\mathcal{F}\cup\{\chi_{0}\}$. This works also for exhaustive families since $\mathcal{F}$ is exhaustive for $A$ if, and only if, $\mathcal{F}^{+}$ is exhaustive for $A^{+}$. Sometimes it is convenient to use the following alternative characterization of strictly spectral sets of representations. The set of representations $\mathcal{F}$ of $A$ is strictly spectral if it satisfies the following property: $1+a\in A^{+}$, $a\in A$, is invertible if, and only if, $1+\phi(a)$ is invertible for any $\phi\in\mathcal{F}$.

The set of all equivalence classes of irreducible representations of a unital $C^{\ast}$-algebra is strictly norming (see [26, 27]). Therefore any exhaustive family is strictly norming [72]. We recall the following results from [72, 84], which establish the relations between these notions.

Theorem 3.3**.**

Let $\mathcal{F}$ be a set of non-degenerate representations of a $C^{\ast}$-algebra $A$. Then $\mathcal{F}$ is strictly norming if, and only if, it is strictly spectral. Every exhaustive set of representations is strictly spectral. If $A$ is furthermore separable, then the converse is also true.

Let $A$ be a $C^{\ast}$-algebra and $I\subset A$ be a closed two-sided ideal. Recall that any nondegenerate representation $\pi\colon I\to\mathcal{L}(\mathcal{H})$ extends to a unique representation $\pi\colon A\to\mathcal{L}(\mathcal{H})$. (See [26, Proposition 2.10.4].) This leads to the following results (see [72], Proposition 3.15 and Corollary 3.16).

Proposition 3.4**.**

Let $I\subset A$ be an ideal of a $C^{\ast}$-algebra. Let $\mathcal{F}_{I}$ be a set of nondegenerate representations of $I$ and $\mathcal{F}_{A/I}$ be a set of representations of $A/I$. Let $\mathcal{F}:=\mathcal{F}_{I}\cup\mathcal{F}_{A/I}$, regarded as a family of representations of $A$. If $\mathcal{F}_{I}$ and $\mathcal{F}_{A/I}$ are both exhaustive, then $\mathcal{F}$ is also exhaustive. The same result holds by replacing exhaustive with strictly norming.

In the following corollary, one should think of the invertibility of $a$ in $A/I$ as an “ellipticity” condition.

Corollary 3.5**.**

Let $I\subset A$ be an ideal of a unital $C^{\ast}$-algebra $A$ and let $\mathcal{F}_{I}$ be a strictly spectral set of nondegenerate representations of $I$. Let $a\in A$. Then $a$ is invertible in $A$ if, and only if, it is invertible in $A/I$ and $\phi(a)$ is invertible for all $\phi\in\mathcal{F}_{I}$.

We now address Morita equivalence [83] in a very simple form (see also [75, 81]). It is known that there exists a homeomorphism between the primitive ideal spectra of Morita equivalent $C^{*}$-algebras, so exhaustive families of representations will correspond to exhaustive families of representations under Morita equivalence. In particular, we have the following result.

Proposition 3.6**.**

Let $\mathcal{F}$ be a set of representations of a $C^{*}$-algebra $A$.

1. (i)

If $\mathcal{F}_{n}:=\{\pi\otimes 1|\,\pi\in\mathcal{F}\}$ is the corresponding family of representations of $M_{n}(A)=A\otimes M_{n}(\mathbb{C})$, then $\mathcal{F}$ is exhaustive if, and only if, $\mathcal{F}_{n}$ is exhaustive. 2. (ii)

Let $I\subset A$ be a closed, two-sided ideal. If $\mathcal{F}$ is exhaustive, then $\mathcal{F}_{I}:=\{\pi|_{I}|\,\pi\in\mathcal{F}\}$ is an exhaustive family of representations of $I$. 3. (iii)

Let $e\in M_{n}(A)$ be a projection. Then $\mathcal{F}_{n}$ defines, by restriction, an exhaustive family of representations of $eM_{n}(A)e$.

Proof.

The first part follows by Morita equivalence. The second part follows since $\operatorname{supp}(\phi|_{I})=\operatorname{supp}(\phi)\cap\operatorname{Prim}(I)$. The last part is proved by combining the first two parts. Indeed, $\mathcal{F}_{n}$ defines an exhaustive set of representations of $M_{n}(A)$, and hence an exhaustive set of representations of $M_{n}(A)eM_{n}(A)$. Since the algebra $eM_{n}(A)e$ is Morita equivalent to $M_{n}(A)eM_{n}(A)$, by Proposition 4.27 in [31] (see also [83]), the result follows. ∎

3.2. Exel’s question and properties

We will let $\mathcal{G}$ denote a Lie groupoid and let $\mathcal{R}(\mathcal{G})$ will denote its set of regular representations. The discussion below usually makes sense in the more general setting of locally compact groupoids, but we restrict, nevertheless to Lie groupoids, for simplicity.

In [27], Exel has asked for which groupoids $\mathcal{G}$ is the set $\mathcal{R}(\mathcal{G})$ a strictly norming set of representations of $C_{r}^{\ast}(\mathcal{G})$. An example due to Voiculescu, see [72], shows that not every locally compact groupoid has this property. (The question whether all Lie groupoids have Exel’s property is still open.) It is easier to work with exhaustive families of representations, so we introduce the following definition.

Definition 3.7**.**

We say that $\mathcal{G}$ has Exel’s property if $\mathcal{R}(\mathcal{G})$, the set of regular representations of $\mathcal{G}$, is an exhaustive set of representations for $C_{r}^{\ast}(\mathcal{G})$. We say that $\mathcal{G}$ has Exel’s strong property if $\mathcal{R}(\mathcal{G})$ is an exhaustive set of representations for $C^{\ast}(\mathcal{G})$.

We have that

[TABLE]

Hence, if $\mathcal{F}$ is an exhaustive family of representations of a $C^{*}$-algebra $A$, then $\bigcap_{\phi\in\mathcal{F}}\,\operatorname{Prim}(\ker(\phi))=\emptyset,$ and hence $\mathcal{F}$ is a faithful family of representations of $A$. In particular, $\mathcal{G}$ has Exel’s strong property if, and only if, $\mathcal{G}$ has Exel’s property and is metrically amenable.

We shall need the following consequence of Propositions 2.6 and 3.4 of the previous subsection.

Corollary 3.8**.**

Let $U\subset M$ be an open, $\mathcal{G}$-invariant subset and assume $\mathcal{G}$ to be second countable. We let $F:=M\smallsetminus U$. Let us assume that $\mathcal{G}_{U}$ and $\mathcal{G}_{F}$ have Exel’s strong property. Then $\mathcal{G}$ also has Exel’s strong property.

Proof.

Let $I:=C^{\ast}(\mathcal{G}_{U})=C_{r}^{\ast}(\mathcal{G}_{U})$, $A:=C^{\ast}(\mathcal{G})$. Then $A/I\simeq C^{\ast}(\mathcal{G}_{F})\simeq C_{r}^{\ast}(\mathcal{G}_{F})$ by Proposition 2.6(ii). Since $\mathcal{G}_{F}$ and $\mathcal{G}_{U}$ are metrically amenable, it follows that $\mathcal{G}$ is metrically amenable as well, by Proposition 2.6(iii). Proposition 3.4 then gives that $\mathcal{R}(\mathcal{G})=\mathcal{R}(\mathcal{G}_{U})\cup\mathcal{R}(\mathcal{G}_{F})$ is an exhaustive family of representations of $C^{\ast}(\mathcal{G})\simeq C_{r}^{\ast}(\mathcal{G})$, that is, $\mathcal{G}$ has Exel’s strong property, as claimed. ∎

More generally, we can consider groupoids given by a filtration.

Corollary 3.9**.**

Let $\mathcal{G}\rightrightarrows M$ be a second countable groupoid and assume that $U_{i}$ are open, $\mathcal{G}$-invariant subsets of $M$, such that $\emptyset\,:=\,U_{-1}\subset U_{0}\subset\ldots\subset U_{i-1}\subset U_{i}\subset\ldots\subset U_{N}:=M\,,$ and for each $S:=U_{i}\smallsetminus U_{i-1}$, $\mathcal{G}_{S}$ has Exel’s strong property. Then $\mathcal{G}$ also has Exel’s strong property.

Proof.

Corollary 3.8 yields the result by induction, as follows. If $N=1$, it reduces to Corollary 3.8. Assuming that the result holds for a filtration in $N-1$ sets, then $\mathcal{G}_{U_{N-1}}$ has Exel’s strong property. Since by assumption $\mathcal{G}_{M\setminus U_{N-1}}$ also has, the result follows again by Corollary 3.8. ∎

Here is a simple, but very important example of a class of Lie groupoids $\mathcal{G}$ for which the set $\mathcal{R}(\mathcal{G})$ of regular representations is an exhaustive set of representations of $C^{\ast}(\mathcal{G})$. The idea of the proof below is that, locally, the $C^{*}$-algebra of $\mathcal{G}$ is of the form $C^{*}(G)\otimes\mathcal{K}$, where $\mathcal{K}$ are the compact operators on the (typical) fiber of $f$. We found it convenient to formalize the proof of this well-known result using Morita equivalence of groupoids.

Proposition 3.10**.**

Let $\mathcal{H}\rightrightarrows L$ be a bundle of Lie groups with fiber $G$ and let $f:M\to L$ be a tame submersion. Define $\mathcal{G}:=f^{\downarrow\downarrow}(\mathcal{H})$. Then $\mathcal{G}$ is a Lie groupoid with Exel’s property. If $G$ is amenable, then $\mathcal{G}$ has Exel’s strong property.

Proof.

We have already discussed that $\mathcal{G}:=f^{\downarrow\downarrow}(\mathcal{H})$ is a Lie groupoid (because $f$ is a tame submersion). We endow $\mathcal{G}$ with one of the standard Haar measures coming from the Lie groupoid structure. Let

[TABLE]

Then $X$ defines a Morita equivalence between $\mathcal{G}$ and $\mathcal{H}$, and hence a Morita equivalence between their full and reduced $C^{*}$-algebras of these groupoids, by the classical results of [64, 96]. See also [100]. In particular, $\operatorname{Prim}(C^{\ast}(\mathcal{G}))$ and $\operatorname{Prim}(C^{\ast}(\mathcal{H}))$ are homeomorphic by natural homeomorphisms [83]. The same is true for $\operatorname{Prim}(C_{r}^{\ast}(\mathcal{G}))$ and $\operatorname{Prim}(C_{r}^{\ast}(\mathcal{H}))$.

All irreducible representations of $C_{r}^{\ast}(\mathcal{H})$ factor through an evaluation morphism $e_{x}:C_{r}^{\ast}(\mathcal{H})\to C_{r}^{\ast}(G_{x})\simeq C_{r}^{\ast}(G)$, for some $x\in M$, where $G_{x}$ is the fiber of $\mathcal{H}\to M$ above $x$. We have that the regular representation $\pi_{x}$ is obtained from the regular representation of $G_{x}$ via $e_{x}$, the evaluation at $x$. Hence $\mathcal{G}$ has Exel’s property.

Let us assume now that $G$ is amenable. Hence the fibers of $\mathcal{H}\to M$ are amenable, and therefore $\mathcal{H}$ is metrically amenable. Hence $\mathcal{G}$ is also metrically amenable. Since $\mathcal{G}$ was already proved to have Exel’s property, it follows that it has also Exel’s strong property. ∎

We notice that if $\mathcal{G}$ is (isomorphic to a groupoid) as in the proposition above, then $\mathcal{H}\rightrightarrows L$ and $f:M\to L$ are uniquely determined, up to equivalence. Indeed, $L$ is diffeomorphic to the set of orbits of $\mathcal{G}$ (acting on $M$), $f$ becomes the quotient map $M\to M/\mathcal{G}$ and $\mathcal{H}$ is obtained from the isotropy groupoid of $\mathcal{G}$.

The above results suggest the following definition.

Definition 3.11**.**

Let $\mathcal{H}$ be a groupoid with units $M$. We say that $\mathcal{H}$ is a stratified submersion groupoid with filtration $(U_{i})$ if $U_{i}$ are open, $\mathcal{H}$-invariant subsets of $M$,

[TABLE]

and each $S:=U_{i}\smallsetminus U_{i-1}$ is a manifold, possibly with corners, such that there exist a Lie group bundle $G_{S}\to B_{S}$ and a tame submersion $f_{S}:S\to B_{S}$ of manifolds with corners such that

[TABLE]

A groupoid $\mathcal{H}$ will be called a stratified submersion groupoid if it is a stratified submersion groupoid for some filtration.

Remark 3.12*.*

It is clear from the definition that if $\mathcal{H}$ is a stratified submersion groupoid, then, with the notation of the definition, each $\mathcal{H}_{U_{k}\smallsetminus U_{j}}$, $j<k$, is also a stratified submersion groupoid. We have to notice, however, that $\mathcal{H}_{U_{k}\smallsetminus U_{j}}$ may not be a Lie groupoid, even if $\mathcal{H}$ is a Lie groupoid, because $U_{k}\smallsetminus U_{j}$ may not be a manifold (even if it is a union of manifolds).

As we shall see, the class of stratified submersion groupoids is large enough to comprise many classes of groupoids and operators that arise in practice. Combining Corollary 3.9 with Proposition 3.10, we now obtain the following result.

Theorem 3.13**.**

Let $\mathcal{G}\rightrightarrows M$ be a stratified submersion Lie groupoid and assume that all the isotropy groups $\mathcal{G}_{x}^{x}$, $x\in M$, are amenable. Then the set $\mathcal{R}(\mathcal{G})$ of regular representations of $\mathcal{G}$ is an exhaustive family of representations of $C^{\ast}(\mathcal{G})$. In particular, $\mathcal{G}$ is metrically amenable and has Exel’s property.

Proof.

Let us use the notation of Definition 3.11. Indeed then, the isotropy group $\mathcal{G}_{x}^{x}$ is the fiber of $G_{S}\to B_{S}$ above $f_{S}(x)$, for $x\in S$. Thus, by Proposition 3.10, all the groupoids $\mathcal{G}_{S}$ are metrically amenable and have Exel’s property. Using Corollary 3.9, we obtain that $\mathcal{G}$ has Exel’s strong property, that is, that it is metrically amenable and has Exel’s property. ∎

In the next section, we will see how this property plays a role in establishing Fredholm conditions for operators on groupoids.

Remark 3.14*.*

Let $\mathcal{H}$ be a stratified submersion Lie groupoid as in Definition 3.11. In view of the remark 2.7, we have then the following disjoint union decomposition:

[TABLE]

where $\times_{B}$ is the fibered product over the base (that is, over $B_{S}$). We notice also that two regular representations $\pi_{x}$ and $\pi_{y}$ are unitarily equivalent if there is $g\in\mathcal{G}$ such that $d(g)=x$ and $r(g)=y$. For a stratified submersion groupoid, this is then the case exactly if $x,y\in S:=U_{i-1}\smallsetminus U_{i}$ and $f_{S}(x)=f_{S}(y)=b\in B_{s}$, in which case, these two representations act, up to a unitary equivalence, on the set of square integrable functions on the space $Z_{b}:=f_{S}^{-1}(b)\times(G_{S})_{b}$. We denote by $\pi_{b}$ the corresponding representation on $L^{2}(Z_{b})$, $b\in B_{S}$.

4. Fredholm conditions

We now study Fredholm conditions for operators in algebras $\mathbf{\Psi}$ containing a reduced groupoid $C^{\ast}$-algebra $C^{\ast}_{r}(\mathcal{G})$ as an essential ideal. (Recall that an ideal of an algebra is essential if its annihilator vanishes.) The groupoids for which we obtain the kind of Fredholm conditions that we want (the kind that are typically used in practice) will be called “Fredholm groupoids.” They are introduced and discussed next. Examples of Fredholm groupoids will be provided in the next section.

Throughout the rest of this paper, $\mathcal{G}\rightrightarrows M$ will denote a second countable Lie groupoid, which is hence a locally compact groupoid and the choice of a metric on $A(\mathcal{G})$ gives rise to a Haar system $\lambda_{x}$. Also, recall that all our Lie groupoids are assumed to be Hausdorff.

4.1. Fredholm groupoids and their characterization

We now introduce Fredholm groupoids and give a first characterization of these groupoids. Recall that $\pi_{x}\colon C^{\ast}(\mathcal{G})\to\mathcal{L}(L^{2}(\mathcal{G}^{x},\lambda_{x}))$ denotes be the regular representation on $\mathcal{G}_{x}$, given by left convolution, $\pi_{x}(\phi)\psi:=\phi*\psi$ (Definition 2.4). We shall use the following notation throughout the rest of the paper.

Notations 4.1**.**

Let us assume that $U\subset M$ is an open, $\mathcal{G}$-invariant subset with $\mathcal{G}_{U}\simeq U\times U$ (the pair groupoid, see Example 2.17). For any $x_{0}\in U$, the range map then defines a bijection $r:\mathcal{G}_{x_{0}}\to U$ and hence a measure $\mu$ on $U$ corresponding to $\lambda_{x_{0}}$. This measure does not depend on the choice of $x_{0}\in U$ and leads to isometries $L^{2}(\mathcal{G}_{x_{0}},\lambda_{x_{0}})\simeq L^{2}(U;\mu)$ that commute with the action of $\mathcal{G}$. We then denote by $\pi_{0}$ the corresponding representation of $C^{\ast}(\mathcal{G})$ on $L^{2}(U;\mu)$. It is often called the vector representation of $C^{\ast}(\mathcal{G})$. We shall usually write $L^{2}(U):=L^{2}(U;\mu)$. The representation $\pi_{0}$ then defines an isomorphism $C_{r}^{\ast}(\mathcal{G}_{U})\simeq\pi_{0}(C_{r}^{\ast}(\mathcal{G}_{U})=\mathcal{K}$, the algebra of compact operators on $L^{2}(U)$.

Remark 4.2*.*

An important remark is that, since we only consider Hausdorff Lie groupoids, the vector representation $\pi_{0}$ is injective, by a result of Khoshkam and Skandalis [40]. In particular, we have that $1+\pi_{0}(a)$ is invertible in $\pi_{0}(C_{r}^{\ast}(\mathcal{G}))/\mathcal{K}$ if, and only if, $1+a$ is invertible in $C_{r}^{\ast}(\mathcal{G})/C_{r}^{\ast}(\mathcal{G}_{U})$. We shall thus identify $C^{\ast}_{r}(\mathcal{G})$ with its image under $\pi_{0}$, that is, with a class of operators on $L^{2}(U)$, without further comment.

In the following definition, we shall use that $C_{r}^{\ast}(\mathcal{G}_{U})\subset\ker\pi_{x}$, $x\notin U$, by definition, and hence that $\pi_{x}$ descends to the quotient algebra.

Definition 4.3**.**

We say that $\mathcal{G}$ is a Fredholm Lie groupoid if:

1. (i)

There is an open, dense, $\mathcal{G}$-invariant subset $U\subset M$ such that $\mathcal{G}_{U}\simeq U\times U$. 2. (ii)

Given $a\in C^{\ast}_{r}(\mathcal{G})$, we have that $1+a$ is Fredholm if, and only if, all $1+\pi_{x}(a)$, $x\in F\smallsetminus U$, are invertible (see 4.1 for notation).

The set $F:=M\smallsetminus U$ will be called the set of boundary units of $\mathcal{G}$ and a set $U$ as in this definition will be called a manifold with amenable ends.

For a Fredholm Lie groupoid $\mathcal{G}\rightrightarrows M$, we shall always denote by $U\subset M$ the $\mathcal{G}$-invariant open subset from the definition of a Lie Fredholm groupoid. Since $U$ is a dense orbit, it is uniquely determined by $\mathcal{G}$. Hence, $F:=M\smallsetminus U$ is closed and $\mathcal{G}$-invariant.

We shall need the following result in the matricial case.

Proposition 4.4**.**

Let us assume that $\mathcal{G}$ is a Fredholm Lie groupoid, that $M$ is compact, and that $e\in M_{n}(\mathcal{C}(M))$ is a projection.

1. (i)

The set of representations $\pi_{x}$, $x\in F$ defines a strictly spectral family of representations of eM_{n}\big{(}C_{r}^{\ast}(\mathcal{G})/C_{r}^{\ast}(\mathcal{G}_{U})\big{)}e. 2. (ii)

For any $a\in eM_{n}(C^{\ast}_{r}(\mathcal{G}))e$, we have that $1+\pi_{0}(a)$ is Fredholm if, and only if, all $1+\pi_{x}(a)$, $x\in F:=M\smallsetminus U$, are invertible (see 4.1 for notation).

Proof.

By Atkinson’s theorem (which states that an operator is Fredholm if, and only if, it is invertible modulo the compacts) and by the definition of strictly spectral families of representations in the non-unital case, one can see that condition (ii) in the definition of Fredhom groupoid is equivalent to the fact that $\mathcal{R}(\mathcal{G}_{F})$ defines a strictly spectral family of representations of the quotient $C_{r}^{\ast}(\mathcal{G})/\mathcal{K}\simeq C_{r}^{\ast}(\mathcal{G})/C_{r}^{\ast}(\mathcal{G}_{U})$. Since $\mathcal{G}$ is second countable, its associated $C^{*}$-algebras are separable, hence the set $\{\pi_{x}|x\in F\}$ is strictly spectral if, and only if, it is exhaustive, by Theorem 3.3. The first part is then a consequence of Proposition 3.6.

The second part follows again from the definition of strictly spectral families of representations in the non-unital case, from the fact that $C_{r}^{\ast}(\mathcal{G}_{U})\simeq\mathcal{K}(L^{2}(U))$, and from Atkinson’s theorem. ∎

In the next results, we give an abstract characterization of Lie Fredholm groupoids, using some of the ideas of the previous section. We obtain in particular conditions that will make it simpler to check whether a given Lie groupoid is Fredholm. Recall that, in this paper, all groupoids are Hausdorff.

Theorem 4.5**.**

Assume $\mathcal{G}$ is a Lie groupoid. If $\mathcal{G}$ is Fredholm, then following two conditions are satisfied:

1. (i)

The canonical projection $C_{r}^{\ast}(\mathcal{G})\to C_{r}^{\ast}(\mathcal{G}_{F})$ induces an isomorphism

[TABLE] 2. (ii)

$\mathcal{G}_{F}$* has Exel’s property.*

(Recall that condition (ii) means that $\mathcal{R}(\mathcal{G}_{F})$ is an exhaustive set of representations of $C_{r}^{\ast}(\mathcal{G}_{F})$.)

Proof.

Let us assume that $\mathcal{G}$ is a Fredholm groupoid and check the two conditions of the statement.

Let $p\colon C_{r}^{\ast}(\mathcal{G})\to C_{r}^{\ast}(\mathcal{G}_{F})$ be the natural projection induced by restriction. It is known that $C_{r}^{\ast}(\mathcal{G}_{U})\subseteq\ker(p)$ (see the proof of [81, Prop.II.4.5 (a)]). To prove (i), we need to show that we have equality $C_{r}^{\ast}(\mathcal{G}_{U})=\ker(p)$. Let us proceed by contradiction, that is, let us assume that $C_{r}^{\ast}(\mathcal{G}_{U})\neq\ker(p)$. Then we can choose $a=a^{*}\in\ker(p)\smallsetminus C_{r}^{\ast}(\mathcal{G}_{U})$. Note that since $\pi_{x}=\pi_{x}\circ p$, $x\in F=M\setminus U$, we have $\ker p\subset\ker\pi_{x}$, in particular, $1-\pi_{x}(a)=1$ is invertible for all $x\notin U$. On the other hand, since $a$ is self-adjoint and non-zero, in the quotient $\ker(p)/C_{r}^{\ast}(\mathcal{G}_{U})$, there is $0\neq\lambda\in\mathbb{R}$ such that $\lambda-a$ is not invertible in $\ker(p)/C_{r}^{\ast}(\mathcal{G}_{U})$. By rescaling, we may assume $\lambda=1$ and thus $1-a$ is not invertible in $C^{\ast}_{r}(\mathcal{G})/C_{r}^{\ast}(\mathcal{G}_{U})\cong C^{\ast}_{r}(\mathcal{G})/\mathcal{K}$. We conclude that in this case $\mathcal{R}(\mathcal{G}_{F})$ is not a strictly spectral family of representations of $C_{r}^{\ast}(\mathcal{G})/C_{r}^{\ast}(\mathcal{G}_{U})$. It then follows that $1-a$ is not Fredholm, however $1-\pi_{x}(a)=1$ is invertible for all $x\notin U$. This is a contradiction.

The first part shows that there is an isomorphism

[TABLE]

By Fredholmness of $\mathcal{G}$, we have that $\mathcal{R}(\mathcal{G}_{F})$ defines a strictly spectral family of representations of $C^{\ast}_{r}(\mathcal{G})/\mathcal{K}$, so $\mathcal{R}(\mathcal{G}_{F})$ is also a strictly spectral family of representations of $C_{r}^{\ast}(\mathcal{G}_{F})$. Since $\mathcal{G}$ is second countable, it is exhaustive, by Theorem 3.3, so that $\mathcal{G}_{F}$ has Exel’s property. This proves (ii) and hence the direct implication. ∎

We remark that we proved in fact that, if $\mathcal{R}(\mathcal{G}_{F})$ defines a strictly spectral family of representations of $C_{r}^{\ast}(\mathcal{G})/C_{r}^{\ast}(\mathcal{G}_{U})$, then necessarily $C_{r}^{\ast}(\mathcal{G})/C_{r}^{\ast}(\mathcal{G}_{U})\,\simeq\,C_{r}^{\ast}(\mathcal{G}_{F})$ and $\mathcal{G}_{F}$ has Exel’s property.

A stronger form of the converse of Theorem 4.5 is contained in the following theorem.

Theorem 4.6**.**

Let $\mathcal{G}\rightrightarrows M$ be a Lie groupoid. Let us assume that $U\subset M$ is a dense $\mathcal{G}$-invariant open subset such that $\mathcal{G}_{U}\simeq U\times U$. Also, let us assume that the two conditions of Theorem 4.5 are satisfied. Then, for any unital $C^{\ast}$-algebra $\mathbf{\Psi}$ containing $M_{n}(C^{\ast}_{r}(\mathcal{G}))$ as an essential ideal and for any $a\in\mathbf{\Psi}$, we have that $\pi_{0}(a)$ if Fredholm if, and only if, the image of $a$ in $\mathbf{\Psi}/M_{n}(C^{\ast}_{r}(\mathcal{G}))$ is invertible and all $\pi_{x}(a)$, $x\in M\smallsetminus U$, are invertible.

Proof.

The proof is the same for any $n$ (using also Proposition 3.6, so we let $n=1$ for notational simplicity). Let us assume that conditions (i) and (ii) of Theorem 4.5 are satisfied and let $\mathbf{\Psi}$ be a $C^{\ast}$-algebra containing $C^{\ast}_{r}(\mathcal{G})$ as an essential ideal. Property (i) implies that $\pi_{0}$ is injective on $\mathbf{\Psi}$ since $C^{\ast}(\mathcal{G})$ is an essential ideal of $\mathbf{\Psi}$. By our assumptions on $\mathcal{G}$, we obtain that the algebra $\Psi/C_{r}^{\ast}(\mathcal{G}_{U})\simeq\pi_{0}(\Psi)/\mathcal{K}=:B$ contains $B_{0}:=\pi_{0}(C_{r}^{\ast}(\mathcal{G}))/\mathcal{K}$ as an ideal and $B/B_{0}\simeq\mathbf{\Psi}/C_{r}^{\ast}(\mathcal{G})$. Moreover, $B_{0}\simeq C_{r}^{\ast}(\mathcal{G}_{F})$ by (ii).

Let now $a\in\mathbf{\Psi}$ be arbitrary. By Atkinson’s Theorem, we know that $\pi_{0}(a)$ is Fredholm if, and only if, its image in $B$ is invertible. But by (iii) and by Corollary 3.5, $c\in B$ is invertible if, and only if, the image of $c$ in $\mathbf{\Psi}/C_{r}^{\ast}(\mathcal{G})$ and all $\pi_{x}(c)$ are invertible for all $x\in F$. ∎

Corollary 4.7**.**

Let $\mathcal{G}\rightrightarrows M$ be a Lie groupoid and $\emptyset\neq U\subset M$ be a dense, $\mathcal{G}$-invariant, open subset such that $\mathcal{G}_{U}\simeq U\times U$, and $F=M\setminus U$. Then $\mathcal{G}$ is Fredholm if, and only if, $C_{r}^{\ast}(\mathcal{G})/C_{r}^{\ast}(\mathcal{G}_{U})\simeq\,C_{r}^{\ast}(\mathcal{G}_{F})$ and $\mathcal{R}(\mathcal{G}_{F})$ is a strictly spectral set of representations of $\mathcal{G}_{F}$ (that is, if $\mathcal{G}_{F}$ has Exel’s property).

Recalling the exact sequence in Proposition 2.6, it follows that, in particular, if $\mathcal{G}$ is metrically amenable, then $\mathcal{G}$ is Fredholdm if, and only if, $\mathcal{G}_{F}$ has Exel’s property.

Remark 4.8*.*

It is immediate to observe that, if the set $\mathcal{R}(\mathcal{G}_{F})$ of regular representations $\pi_{x}$, $x\in F$, forms a strictly spectral set of representations of $C^{\ast}(\mathcal{G}_{F})$, then $\mathcal{G}_{F}$ has Exel’s strong property, and conditions (ii) and (iii) of the Theorem 4.5 are satisfied (using Proposition 2.6).

This remark and the above results then give the following sufficient conditions for a Lie groupoid to be Fredholm.

Corollary 4.9**.**

Let $\mathcal{G}\rightrightarrows M$ be a Lie groupoid and $U\subset M$ be a dense $\mathcal{G}$-invariant open subset such that $\mathcal{G}_{U}\simeq U\times U$, and $F=M\setminus U$. Assume that $\mathcal{R}(\mathcal{G}_{F})$ forms a strictly spectral set of representations of $C^{\ast}(\mathcal{G}_{F})$ or, equivalently, that $\mathcal{G}_{F}$ has Exel’s strong property. Then $\mathcal{G}$ is a Fredholm groupoid.

See also [7]. Together with Theorem 3.13, we obtain an important class of Fredholm groupoids. Recall that we assume that all our groupoids are Hausdorff.

Theorem 4.10**.**

Let $\mathcal{G}\rightrightarrows M$ be a stratified submersion Lie groupoid with filtration $U_{i}$ such that $U_{0}$ is dense and $\mathcal{G}_{U_{0}}\simeq U_{0}\times U_{0}$. Assume that all isotropy groups $\mathcal{G}_{x}^{x}$ are amenable. Then $\mathcal{G}$ is a Fredholm groupoid.

Proof.

By Remark 3.12 applied to $F:=M\smallsetminus U_{0}$ and by Theorem 3.13, we know that the family $\mathcal{R}(\mathcal{G}_{F})$ of regular representations of $\mathcal{G}_{F}$ is an exhaustive family of representations of $C^{\ast}(\mathcal{G}_{F})$, that is, $\mathcal{G}_{F}$ has Exel’s strong property. The result follows from the previous corollary. ∎

We notice that we can replace $\pi_{x}$, $x\in F$, with $\pi_{b}$, $b\in B_{S}$, for the strata corresponding to the boundary. We can further restrict to the set of $b$s in a dense set of orbits, since they give rise to a faithful family of representations of $\mathcal{G}_{F}$, again by [40].

4.2. Pseudodifferential operators

So far we have discussed mostly groupoids and operators in their (unitized) $C^{\ast}$-algebras. The study of differential operators on suitable non-compact spaces can be reduced to the study of groupoid algebras using pseudodifferential operators on Lie groupoids. In this subsection, we will explain this reduction procedure. See [38, 68, 87, 95, 105] for introductions to pseudodifferential operators.

Let $\mathcal{G}$ be a Lie groupoid and consider the algebra $\Psi^{*}(\mathcal{G})$ of pseudodifferential operators on the groupoid $\mathcal{G}$, whose definition we now briefly recall [4, 61, 73]. Then, for $m\in\mathbb{Z}\cup\{\pm\infty\}$, $\Psi^{m}(\mathcal{G})$ consists of smooth families $(P_{x})_{x\in M}$ of classical pseudo-differential operators $P_{x}\in\Psi^{m}(\mathcal{G}_{x})$ of order $m$, that are right invariant with respect to the action of $\mathcal{G}$ and have compactly supported distribution kernels. In particular, $\Psi^{-\infty}(\mathcal{G})$ is nothing but the convolution algebra of smooth, compactly supported function on $\mathcal{G}$, that is, $\Psi^{-\infty}(\mathcal{G})\simeq{\mathcal{C}}^{\infty}_{\text{c}}(\mathcal{G})$.

This construction extends right away to operators between smooth sections of some vector bundles $E,F\to M$. There are two methods of doing this. The first method is to consider families $(P_{x})_{x\in M}$ of operators $P_{x}\in\Psi^{m}(\mathcal{G}_{x};r^{*}(E),r^{*}(F))$. We denote by $\Psi^{m}(\mathcal{G};E,F)$ the resulting set of operators. By embedding $E\oplus F$ into a trivial bundle of dimension $N$, we can identify $\Psi^{m}(\mathcal{G};E,F)$ with a subspace of $M_{N}(\Psi^{m}(\mathcal{G}))$, when convenient. Similar considerations yield differential operators generated by $\mathcal{V}$ acting between vector bundles $E$ and $F$ endowed with metric preserving connections.

The second method to deal with operators between two vector bundles $E,F\to M$ is as follows. Let us choose an embedding of $E\oplus F\to\underline{\mathbb{C}}^{N}$ into a trivial vector bundle and denote by $e$ and $f$ the corresponding orthogonal projections onto the images of $e$ and $f$. Then we can identify

[TABLE]

This idea applies to other types of similar operators (norm closures of operators in $\Psi^{m}(\mathcal{G})$, differential operators, … ).

One of the main points of considering pseudodifferential operators on groupoid is that, if we denote $\mathcal{V}:=\operatorname{Lie}(\mathcal{G}):=\varrho(\Gamma(A(\mathcal{G})))$, then

[TABLE]

where $\operatorname{Diff}^{m}(\mathcal{V};E,F)$ is generated by ${\mathcal{C}}^{\infty}(M;\operatorname{End}(E\oplus F))$ and $\mathcal{V}$ (using compatible connections on $E$ and $F$) and $\operatorname{Diff}(M;E,F)$ denotes all differential operators on $M$ acting between sections of $E$ and $F$. See also 4.14.

Another main point for introducing the algebra $\Psi^{*}(\mathcal{G})$ is related to parametrices and inverses of differential operators. To discuss this, let us denote by $\overline{\Psi}(\mathcal{G})$ the $C^{\ast}$-algebra obtained as the closure of $\Psi^{*}(\mathcal{G})$ with respect to all contractive $*$-representations of $\Psi^{-\infty}(\mathcal{G})$, as in [45]. As we will see below, this definition extends to operators acting between vector bundles. Let us denote, as usual, by $S^{*}A$ the set of unit vectors in the Lie algebroid $A^{*}(\mathcal{G})$ associated with $\mathcal{G}$ (as in Remark 2.11), with respect to some fixed metric on $A^{*}(\mathcal{G})$. Then $\overline{\Psi}(\mathcal{G})$ fits into the following exact sequence

[TABLE]

(See for instance [44] and the references therein.) Moreover, if $P\in\Psi^{m}(\mathcal{G};E,F)$, $m\geq 0$, is elliptic and invertible in $L^{2}$, then its inverse will be in $\overline{\Psi}(\mathcal{G};F,E)$. Typically, Fredholm conditions are obtained for Fredholm Lie groupoids by applying our results to the algebra $\mathbf{\Psi}=M_{N}(\overline{\Psi}(\mathcal{G}))$, for some $N$. Let us see how this is done.

We first need to recall the definition of Sobolev spaces. Let us fix in this subsection a Lie groupoid $\mathcal{G}\rightrightarrows M$. For the purpose of the next result, let us assume that its space of units $M$ has an open, dense, $\mathcal{G}$-invariant subset $U\subset M_{0}:=M\smallsetminus{\partial}M$ such that the restriction $\mathcal{G}_{U}$ is isomorphic to the pair groupoid $U\times U$. Let us also assume that the space of units $M$ is compact. This is needed in order to construct canonical Sobolev spaces on the interior of $M$. Indeed, there is an essentially unique class of metrics on $A(\mathcal{G})$, which, by restriction, gives rise to a class of metrics on $U$, that are called compatible (with the groupoid $\mathcal{G}$), see also [3]. All these metrics are Lipschitz equivalent and complete [2, 45]. In fact, the Sobolev spaces of all these metrics will coincide. They are given as the domains of the powers of $1+\Delta$, where $\Delta$ is the (geometer’ s, i.e. positive) Laplacian. We shall denote by $H^{s}(U)=H^{s}(M)$ these Sobolev spaces. The Sobolev spaces $H^{s}(M)$ are discussed in detail in [2]. See also [71] for a review. The operators in $\Psi^{m}(\mathcal{G})$ (and their vector bundle analogues) model differential operators associated to (any) compatible metric (see also Remark 4.14 below). By considering a vector bundle $E\to M$ and $\Delta_{E}$ the associated Laplacian on sections of $E$, we obtain the spaces $H^{s}(M;E)$.

Note that the Sobolev spaces $H^{s}(M)$ are not defined using the compact manifold structure on $M$, but rather using the complete metric on $U\subset M$. Recall that if the set $M\setminus U$ is as in Definition 4.3, then the set $U$ is called a manifold with amenable ends. If $P\in\Psi^{m}(\mathcal{G};E,F)$, then $P:H^{s}(M;E)\to H^{s-m}(M;F)$ is continuous.

Let $\Psi^{m}(\mathcal{G};E,F)$ denote the pseudodifferential operators acting on the lifts of $E$ and $F$ to $\mathcal{G}$ via $d$. We let $\Psi^{m}(\mathcal{G};E):=\Psi^{m}(\mathcal{G};E,E)$. Then if $P:H^{s}(M;E)\to H^{s-m}(M;F)$ is an order $m$ pseudodifferential operator, we replace the study of its Fredholm properties with those of

[TABLE]

where $Q_{m}$ is an invertible order $m$ pseudodifferential operator acting on the complement of $E\oplus F$. Let $G$ denote the complement of $E\oplus F$ in $\underline{\mathbb{C}}^{N}$. We fix the smooth vector bundles $E,F,G\to M$ in what follows. We denote by $\Delta_{E}$, $\Delta_{F}$, and $\Delta_{G}$ the associated Laplacians acting on sections of $E$, respectively, $F$ and $G$. For instance, $Q_{m}\in\Psi^{m}(\mathcal{G};G)$ could be $(1+\Delta_{G})^{m/2}$ (or a suitable approximation it, see [45]).

One of the main drawbacks of the algebras $\Psi^{m}(\mathcal{G})$ is that they are too small to contain resolvents. This issue is easily fixed by considering completions with respect to suitable norms. Let us consider the norm $\|\,\cdot\,\|_{m,s}$ defined by.

[TABLE]

We then let

[TABLE]

Recall that $H^{s}(M;E)$ is the domain of $(1+\Delta_{E})^{s/2}$, if $s\geq 0$, whenever $M$ is compact (see [2, 33, 45]. Hence $L^{m}_{s}(\mathcal{G};E,F)$ is the norm closure of $\Psi^{m}(\mathcal{G};E,F)$ in the topology of continuous operators $H^{s}(M;E)\to H^{s-m}(M;F)$. We note that, if $P\in\Psi^{m}(\mathcal{G};E,F)$, then

[TABLE]

by [45] (see also [5]). Moreover, let

[TABLE]

Then $\mathcal{W}^{m}(\mathcal{G})\subset L^{m}_{s}(\mathcal{G})$ and $\mathcal{W}^{\infty}(\mathcal{G})$ is an algebra of pseudodifferential operators that contains the inverses of its $L^{2}$-invertible operators.

Recall that we denote by $\pi_{0}:C^{\ast}(\mathcal{G})\to\mathcal{L}(L^{2}(M))$ the vector representation and that $M$ is compact. It is unitarily equivalent to the regular representations $\pi_{x}$, $x\in U$. Moreover, since $\mathcal{G}$ is Hausdorff, $\pi_{0}$ is injective. We have the following result from [45].

Proposition 4.11**.**

Let $\mathcal{G}\rightrightarrows M$ be a Lie groupoid with $\mathcal{G}_{U}=U\times U$ and $M$ compact. Then $M_{N}(\overline{\Psi}(\mathcal{G}))=M_{N}(L_{0}^{0}(\mathcal{G}))$ contains $M_{N}(C^{\ast}(\mathcal{G}))$ as an essential ideal. Let $s\in\mathbb{R}$ and $P\in L^{m}_{s}(\mathcal{G};E,F)\supset\Psi^{m}(\mathcal{G};E,F)$. Then

[TABLE]

We have that $P:H^{s}(M;E)\to H^{s-m}(M;F)$ is Fredholm if, and only if, $a:L^{2}(M;E)\to L^{2}(M;F)$ is Fredholm.

This proposition applies, in particular, if $\mathcal{G}$ is a Fredholm Lie groupoid with $M$ compact.

Proof.

We can assume $N=1$. The fact that $\overline{\Psi}(\mathcal{G})$ contains $C^{\ast}(\mathcal{G})$ as an essential ideal is a general fact–true for any Lie groupoid. This general fact is true because it is true for any non-compact manifold, in particular, for each of the manifolds $\mathcal{G}_{x}$. We have, by the definitions of Sobolev spaces and of Fredholm operators, that $P$ is Fredholm if, and only if, $a$ is Fredholm. The fact that $a:=(1+\Delta_{F})^{(s-m)/2}P(1+\Delta_{E})^{-s/2}\in\overline{\Psi}(\mathcal{G};E,F)$ follows from the definition of $L_{s}^{m}(\mathcal{G};E,F)$, Equation (6) and the results in [45], as noticed already above. ∎

This proposition then gives right away the following result (for technical reasons, we may have to replace $P$ with $a$ in certain proofs). Recall that an operator $P\in\Psi^{m}(\mathcal{G};E,F)$ consists of a right invariant family $P=(P_{x})$, $x\in M$, with $P_{x}$ acting between the sections of $r^{*}(E)$ and $r^{*}(F)$ on $\mathcal{G}_{x}$. For simplicity, we shall drop $r^{*}$ from the notation below, since there is no danger of confusion. We have $P_{x}=\pi_{x}(P)$. The operators $P_{x}$, $x\in M\setminus U$, will be called limit operators (of $P$). They go back to [28].

Theorem 4.12**.**

Let $\mathcal{G}\rightrightarrows M$ be a Fredholm Lie groupoid with $M$ compact and $\emptyset\neq U\subset M$, open such that $\mathcal{G}_{U}=U\times U$. Let $s\in\mathbb{R}$ and $P\in L^{m}_{s}(\mathcal{G};E,F)\supset\Psi^{m}(\mathcal{G};E,F)$. We have

[TABLE]

Proof.

Let us use the notation of Proposition 4.11. By replacing $P$ with $a$ and using Proposition 4.4, we may assume that we work with $N\times N$ matrices of operators. The proof is the same for all $N$, so we let $N=1$. Since $\mathcal{G}$ is Fredholm, Theorem 4.6, applied to $\mathbf{\Psi}:=\overline{\Psi}(\mathcal{G})$, gives that $a\in\overline{\Psi}(\mathcal{G})$ is Fredholm if, and only if, its image in $\overline{\Psi}(\mathcal{G})/C^{\ast}(\mathcal{G})$ is invertible and all the operators $\pi_{x}(a)$ are invertible. We then notice that $\pi_{x}(a)=(1+\Delta_{x})^{(s-m)/2}P_{x}(1+\Delta_{x})^{-s/2}$ since the extension of $\pi_{x}$ to operators affiliated to $\overline{\Psi}(\mathcal{G})$ is given by $\pi_{x}(P)=P_{x}$ since this is true for $P\in\Psi^{0}(\mathcal{G})$ and $\pi_{x}(\Delta)=\Delta_{x}$, the Laplacian on $\mathcal{G}_{x}$ by [45]. ∎

Remark 4.13*.*

To obtain the result as formulated in Theorem 1.1, we let $\alpha$ be the set of orbits of $\mathcal{G}$ acting on $F:=M\smallsetminus U$, $M_{0}:=U$, $M_{\alpha}:=\mathcal{G}_{x}$, for some $x$ in the orbit $\alpha$, $G_{\alpha}:=\mathcal{G}_{x}^{x}$, and $P_{\alpha}:=\pi_{x}(P)$. It may be useful to notice here that often the bundle $M_{\alpha}\to M_{\alpha}/G_{\alpha}$ is trivial. This is the case, for example, for stratified submersion groupoids.

Remark 4.14*.*

Let $\mathcal{V}:=r_{*}(\Gamma(A(\mathcal{G}))$ be the vector fields on $M$ coming from the infinitesimal action of $\mathcal{G}\rightrightarrows M$ on $M$. We denote by $\operatorname{Diff}^{m}(\mathcal{V})$ the set of order $m$ differential operators on $M$ generated by $\mathcal{V}$ and multiplication with functions in ${\mathcal{C}}^{\infty}(M)$ and $\operatorname{Diff}(\mathcal{V})=\cup_{m}\operatorname{Diff}^{m}(\mathcal{V})$. We denote by $\operatorname{Diff}^{m}(\mathcal{V};E,F)$ the analogous differential operators acting between sections of vector bundles $E,F\to M$. We have that all geometric operators (Laplace, Dirac, Hodge, … ) associated to a compatible metric on $M$ (one that comes by restriction from $A(\mathcal{G})$) belong to $\operatorname{Diff}^{m}(\mathcal{V};E,F)$ for suitable $E,F\to M$; moreover, $\pi_{x}(D)$ for a geometric operator is of the same type as $D$. [3].

Remark 4.15*.*

If $\mathcal{G}$ is $d$-connected (in the sense that the fibers of $d:\mathcal{G}\to M$ are connected), then the orbits of the vector fields $\mathcal{V}$ are the same as the orbits of $\mathcal{G}$. If $x\in M$ and $\alpha\subset M$ is its orbit, then this orbit (with its intrinsic manifold topology) is diffeomorphic to the set $M_{\alpha}/G_{\alpha}$. This makes more explicit the data in Remark 4.13.

Although we shall not use this in the present paper, let us record the consequence for essential spectra. Notice that we do not need the closure of the unions for the essential spectra. We denote by $\sigma(Q)$ the spectrum of an operator $Q$, defined as the set of $\lambda\in\mathbb{C}$ such that $Q-\lambda$ is not invertible, and by $\sigma_{\mathrm{ess}}(Q)$ its essential spectrum, defined as the set of $\lambda\in\mathbb{C}$ such that $Q-\lambda$ is not Fredholm. Here $Q$ may be unbounded. If $P\in\Psi^{m}(\mathcal{G};E)$, $m>0$, we consider it to be defined on $H^{m}$.

Corollary 4.16**.**

Consider the framework of Theorem 4.12 and $P\in\Psi^{m}(\mathcal{G};E)$, $m\geq 0$, elliptic. We have

[TABLE]

Thus, in the scalar, order zero case, we have

[TABLE]

Combining Theorems 4.12 and 4.10, we obtain the following result (recall that all our groupoids are Hausdorff and second countable).

Theorem 4.17**.**

Let $M$ be compact and $\mathcal{G}\rightrightarrows M$ be a stratified submersion Lie groupoid with filtration $U_{i}$ such that $U_{0}$ is dense in $M$ and $\mathcal{G}_{U_{0}}\simeq U_{0}\times U_{0}$. Assume that all isotropy groups $\mathcal{G}_{x}^{x}$ are amenable. Then, for any smooth vector bundles $E,F\to M$ and any $P\in L^{m}_{s}(\mathcal{G};E,F)\supset\Psi^{m}(\mathcal{G};E,F)$, we have

[TABLE]

We actually have $P_{x}:H^{s}(\mathcal{G}_{x};r^{*}(E))\to H^{s-m}(\mathcal{G}_{x};r^{*}(F))$, but, as we have explained above, we drop $r^{*}$ from the notation, for the sake of simplicity, since there is no danger of confusion. We will continue to do that in all related statements below.

Remark 4.18*.*

If $\mathcal{G}$ be as in the above theorem (a stratified submersion groupoid with filtration $U_{i}$), and assume that it is also $d$-connected, as in Remark 4.13. Then, when $P=D$ is a differential operator, we can identify the limit operators $D_{\alpha}$ (and hence also $M_{\alpha}$ and $G_{\alpha}$) in a direct way. Indeed, let $\alpha$ be the $\mathcal{V}$-orbit = the $\mathcal{G}$-orbit through some $x\in M\smallsetminus U_{0}$. We have $G_{\alpha}=\mathcal{G}_{x}^{x}$, as discussed in Remark 4.13. Then $M_{\alpha}=\alpha\times G_{\alpha}$. Let $D\in\operatorname{Diff}(\mathcal{V})$. By linearity, we may assume $D=aX_{i_{1}}X_{i_{2}}\ldots X_{i_{m}}$, with $X_{i}\in\mathcal{V}$ a local basis of $\mathcal{V}$ near $x$ and $a\in{\mathcal{C}}^{\infty}(M)$. (We may also assume that $i_{j}$ form a non decreasing sequence, but that is not necessary.) Since $\alpha$ is an orbit, we have that this basis $(X_{j})$ consists of vectors tangent to $\alpha$. We can choose this basis such that $X_{1},\ldots,X_{k}$ form a local basis of $T_{x}\alpha$ and $X_{j}$, $j>k$, are zero on the orbit $\alpha$ (where they are defined). Consequently, the vector fields $X_{j}$, $j>k$, come from a basis $\tilde{X}_{j}$, $j>k$, of $Lie(G_{\alpha})$. We then have

[TABLE]

where $Y_{j}=X_{i_{j}}|_{\alpha}$, if $i_{j}\leq k$, and $Y_{j}=\widetilde{X}_{i_{j}}$ if $i_{j}>k$. We shall say that the variables $Y_{j}$ (or $X_{j}$) with $j>k$ are ghost derivatives at the considered orbit $\alpha$ (see Remark 4.18).

See also [9, 8, 97, 103]. It would be interesting to extend the results of this section to $L^{p}$-spaces in view of [1, 78].

5. Examples

In this section, we use the results of the previous section to obtain Fredholm conditions for operators on some standard non-compact manifolds: manifolds with (poly)cylindrical ends, asymptotically Euclidean manifolds, asymptotically hyperbolic manifolds, boundary fibration structures, and others. We tried to write this section in such a way that it can to a large extent be read independently of the rest of the paper. The reader interested only in applications can start reading the paper with this section.

The general setting of this section is that of a (non-compact) smooth manifold $M_{0}$ whose geometry is determined by a compactification $M$ to a manifold with corners and a Lie algebra of vector fields $\mathcal{V}$ on $M$. (Although we shall not use this, let us mention for people familiar with the concept, that $(M,\mathcal{V})$ will be a Lie manifold with some additional properties.) The differential (and pseudodifferential) operators considered and for which we obtain Fredholm conditions are the ones generated by $\mathcal{V}$ and ${\mathcal{C}}^{\infty}(M)$, that is, the ones in $\operatorname{Diff}(\mathcal{V})$ and its variants for vector bundles. (Recall that $\operatorname{Diff}(\mathcal{V})$ was introduced in Remark 4.14.) The proof of the Fredholm conditions in this section are obtained as particular cases of Theorem 4.12 by showing that appropriate groupoids are Hausdorff stratified submersion Lie groupoids with amenable isotropy groups and hence that they are Fredholm groupoids, in view of Theorem 3.13. Many of the results below can also be obtained from the results in [44, 45], but the approach followed here aims to be more convenient for non-specialists.

In this section, we continue to denote by $\mathcal{G}$ a Lie groupoid with units $M$, a manifold of dimension $n$. In the theorem yielding Fredholm conditions, $M$ will be assumed compact and endowed with a smooth metric $h$. We assume that $h$ is defined everywhere on $M$, in particular, that it extends to a smooth manifold containing $M$ as a submanifold. Also, usually it will be no loss of generality to assume $M$ connected. Our results are formulated for operators in $L_{s}^{m}(\mathcal{G};E,F)$, which is a suitable completion of $\Psi^{m}(\mathcal{G};E,F)$, see Equations (5) and (6). This allows us to greatly enlarge the scope of our results since the completion procedure defining the spaces $L_{s}^{m}$ leads to algebras that are closed under taking the inverses of $L^{2}$-invertible elements.

5.1. Examples related to group actions

We include first some examples that are closely related to group actions.

5.1.1. The action of a group on a space

Let us assume that Lie group $G$ acts smoothly on a manifold $M$. This yields the transformation (or group action) Lie groupoid $\mathcal{G}:=M\rtimes G$, which, as a set, consists of $M\times G$ and has units $Ma$. We have

[TABLE]

If $M=G$ with $G$ acting by translations, then $M\rtimes G\simeq G\times G$, the product groupoid. Therefore, if $G\subset M$ as a dense, $G$-invariant open set, and $G$ acts on itself by left translation, we are in the setting in which we can ask if the resulting groupoid is Fredholm Lie groupoid. This groupoid was used in [30, 51, 63]. The groupoids used in [30, 63] turn out to be stratified submersion groupoids. As discussed in those papers, this recovers the classical HVZ-theorem [20, 80, 99] as a particular case of Theorem 4.12 (with $U=G$). We note that $M\times G$ is always Hausdorff (since $M$ and $G$ are Hausdorff). If $V\subset M$ is an open subset, then the reduction groupoid $(M\rtimes G)_{V}^{V}$ will be called a local transformation (or action) groupoid, and will also be Hausdorff. Many related results (including the non Lie case), were also obtained by Bottcher, Chandler-Wilde, Karlovich, Lindner, Rabinovich, Roch, Rozenblyum, Silberman, and many others. See [6, 10, 11, 15, 55, 77, 78, 79, 85] and the references therein.

This groupoid models pseudodifferential operators compatible with any $G$ invariant metric on $G$. Let $\mathfrak{g}$ be the Lie algebra of $G$. Then $A(\mathcal{G})=M\times\mathfrak{g}$, with $\mathfrak{g}$ acting on $M$ via the infinitesimal action of $G$. The associated Lie algebra of vector fields is $\operatorname{Lie}(\mathcal{G}):=\varrho(\Gamma(A(\mathcal{G})))=\mathcal{V}:={\mathcal{C}}^{\infty}(M)\mathfrak{g}\subset\Gamma(T\Omega)$. We have that $\operatorname{Diff}(\mathcal{V})$ is generated by ${\mathcal{C}}^{\infty}(M)$ and $\mathfrak{g}$.

5.1.2. The $b$-groupoid

Let $M$ be a manifold with corners. Then the $b$-groupoid of [45, 57, 61, 73] is defined as a set as the disjoint union

[TABLE]

where $F$ ranges through the open, connected faces of $M$, $k_{F}$ is the codimension of the face $F$, $F\times F$ is the pair groupoid and $(\mathbb{R}_{+}^{*})^{k_{F}}$ is a group for componentwise multiplication (and hence also a Lie groupoid).

To obtain the smooth structure on this groupoid, we notice that, locally, it is a transformation groupoid $[0,\infty)^{k}\rtimes(\mathbb{R}_{+}^{*})^{k}$. More precisely, let us choose $g\in\mathcal{G}_{b}$. Then $g=(x,y,v)\in(F\times F)\times\mathbb{R}^{k_{F}}$ for some connected open face $F\subset M$. We can choose a coordinate system $V\subset M$ such that $x,y\in V$, $\overline{V}$ is compact in $F$, and we have a tubular neighborhood $V\times[0,\epsilon)^{k_{F}}\subset M$. Then, a neighborhood of $g$ in $\mathcal{G}_{b}$ is diffeomorphic to the local transformation groupoid obtained by reducing to $V\times[0,\epsilon)^{k_{F}}$ the action groupoid

[TABLE]

where $d_{F}=n-k_{F}$ is the dimension of $F$. Here $\mathbb{R}^{d_{F}}$ acts by translations on itself and $\mathbb{R}_{+}^{*}=(0,\infty)$ acts by multiplication on $[0,\infty)$. In applications, it will be, in fact, more convenient to notice that $\mathbb{R}^{d_{F}}\rtimes\mathbb{R}^{d_{F}}$ is the pair groupoid, and hence to identify a neighborhood of $g$ with a reduction of

[TABLE]

the product of the pair groupoid $V\times V$ and the transformation groupoid $[0,\infty)^{k_{F}}\rtimes(0,\infty)^{k_{F}}$.

If $M$ has embedded faces, that is, if each hyperface $H$ has a defining function $r_{H}$, then we can identify $\mathcal{G}_{b}$ with an open subset of Monthubert’s realization of the $b$-groupoid [62, Proposition 4.5]

[TABLE]

where $\mathcal{H}$ denotes the set of hyperfaces of $M$ and $H,K\in\mathcal{H}$. See also [47, 48, 86, 13, 101].

We see that the $b$-groupoid is a stratified submersion gropoid as follows. The set $U_{k}$, $k\leq n$, is defined as the union of the open faces of codimension $k$ of $M$. Then $(\mathcal{G}_{b})_{U_{k}\setminus U_{k-1}}$ is isomorphic to the topological disjoint union of the groupoids $(F\times F)\times\mathbb{R}^{k}$, where $F$ ranges through the set of open faces of codimension $k$. In particular, $(\mathcal{G}_{b})_{U_{k}\setminus U_{k-1}}$ is isomorphic to the fibered pull-back of a bundle of Lie groups, by Example 2.20. In particular, $(\mathcal{G}_{b})_{x}^{x}\cong\mathbb{R}^{n-k}$ is amenable, for all $x\in M$.

5.1.3. Manifolds with poly-cylindrical ends

The Lie algebroid of $\mathcal{G}_{b}$ is identified by

[TABLE]

Let $h$ be an ordinary metric on $M$. The general form of a compatible metric on $M$ is then

[TABLE]

Manifolds with metrics of this form will be called manifolds with poly-cylindrical ends, following [59]. The Lie algebroid $A(\mathcal{G}_{b})$ is often denoted $T^{b}M$. The groupoid $\mathcal{G}_{b}$ models pseudodifferential operators compatible with the metric $g_{b}$. Thus, by definition, the metric $g_{b}$ comes from restriction from the Lie algebroid $A(\mathcal{G}_{b})$, since $A(\mathcal{G}_{b})|_{M_{0}}=TM_{0}$. Since the base $M$ is compact, all metrics on $A(\mathcal{G}_{b})$ will be equivalent, so the constructions will not depend (essentially) on the choice of the metric. In particular, all geometric operators associated to the metric $g_{b}$ (Laplace, Dirac, Hodge, … ) will belong to $\Psi^{m}(\mathcal{G}_{b};E,F)$ (which is independent of the metric), for suitable $E$ and $F$. Moreover, it turns out that Theorem 4.17 applies to $\mathcal{G}_{b}$ and $\Psi^{m}(\mathcal{G}_{b};E,F)$, provided that $M$* is compact.* (See also Remark 4.14. Further details can be found in [3].)

The statement of Theorem 4.5 can be (slightly) simplified in this case by noticing the following. The representations $\pi_{x}$ and $\pi_{y}$ are unitarily equivalent if $x$ and $y$ are in the same open face $F$, in which case, they will act on $F\times\mathbb{R}^{k_{F}}\simeq\mathcal{G}_{x}$. Let $\pi_{F}$ be the associated representation. For Fredholm conditions, it is enough to consider the invertibility of the operators $P_{H}:=\pi_{x}(P)$, $x\in H$, for the faces of maximal dimension (that is, for hyperfaces) in order to obtain Fredholm conditions, since $\pi_{x}$ is weakly contained in $\pi_{y}$ if $x$ is contained in the closure of the face containing $y$. Let $\mathcal{H}$ denote the set of hyperfaces of $M$, as before. Recall that $L^{m}_{s}(\mathcal{G};E,F)$ is a suitable completion of $\Psi^{m}(\mathcal{G};E,F)$, see Equations (5) and (6). We obtain the following result [58].

Corollary 5.1**.**

Let $P\in L_{s}^{m}(\mathcal{G}_{b};E,F)\supset\Psi^{m}(\mathcal{G}_{b};E,F)$, $M$ compact. We have

[TABLE]

All geometric differential operators associated to the metric $g_{b}$ belong to $\operatorname{Diff}^{m}(\mathcal{V}_{b};E,F)\subset\Psi^{m}(\mathcal{G}_{b};E,F)$, for suitable vector bundles $E,F\to M$.

If $P\in\operatorname{Diff}^{m}(\mathcal{V}_{b};E,F)$, the limit operators $P_{x}$, $x\in{\partial}M$, are obtained as explained in Remark 4.18. They are invariant with respect to the action of the isotropy group $\mathcal{G}_{x}^{x}$ (this is always the case for operators of the form $\pi_{x}(a)$). In this example, only derivatives of the form $x_{H}{\partial}_{x_{H}}$ are ghost derivatives (see Remark 4.18).

Remark 5.2*.*

Carvalho and Qiao have constructed in [14] a similar groupoid to the $b$-groupoid in order to study layer potentials. Their groupoid, however, was not $d$-connected, in general. Nevertheless, the above corollary generalizes to their setting, after some obvious modifications.

5.1.4. Asymptotically Euclidean spaces

Let us assume that $M$ has a smooth boundary ${\partial}M$ with defining function $x=x_{{\partial}M}$ and let $\mathcal{V}_{sc}:=r\mathcal{V}_{b}$. The resulting differential operators $\operatorname{Diff}(\mathcal{V}_{sc})$ and the associated pseudodifferential ooperators are the SG-operators of [19, 74, 88, 90, 89] (called “scattering operators” in [59]). They can be obtained by considering the groupoid

[TABLE]

To obtain a manifold structure on $\mathcal{G}_{sc}$, let us consider first $G=\mathbb{R}^{n}$ and $M=$ the radial compactification of $G$ with the induced action of $G$. Then $\mathcal{G}_{sc}=M\rtimes G$ is Hausdorff and a stratified submersion groupoid. In general, $\mathcal{G}_{sc}$ is locally of this form (and can be obtained by glueing reductions of such groupoids), and hence it is Hausdorff. It satisfies $Lie(\mathcal{G}_{sc}):=\Gamma(A(\mathcal{G}_{sc}))=\mathcal{V}_{sc}.$ Thus, if $g_{b}$ denotes a $b$-metric on $M$ (or rather, on $M_{0}$, then the natural metric associated to $\mathcal{V}_{sc}$ is

[TABLE]

where $g_{b}$ is as in Equation (13).

We have that the orbits of $\mathcal{G}$ on $F:=M\smallsetminus U_{0}={\partial}M$ are reduced to points and that each stabilizer $\mathcal{G}_{x}^{x}=T_{x}M\simeq\mathbb{R}^{n}$, for $x\in F$. In particular, $\mathcal{G}_{x}=\mathcal{G}_{x}^{x}$, for $x\in{\partial}M$ and all derivatives at the boundary are ghost derivatives. If $P\in\operatorname{Diff}^{m}(\mathcal{V}_{sc};E,F)$, the limit operators $P_{x}$ are obtained as explained in Remark 4.18. Theorem 4.17 becomes in our case:

Corollary 5.3**.**

Let $P\in L_{s}^{m}(\mathcal{G}_{sc};E,F)\supset\Psi^{m}(\mathcal{G}_{sc};E,F)$, $M$ compact. We have

[TABLE]

All geometric differential operators associated to the metric $g_{sc}$ belong to $\operatorname{Diff}^{m}(\mathcal{V}_{sc};E,F)\subset\Psi^{m}(\mathcal{G}_{sc};E,F)$, for suitable vector bundles $E,F\to M$.

We have $P_{x}=\pi_{x}(P)$ and $P_{x}$ is translation invariant (i.e. constant coefficient, in this case), and hence can be studied using the Fourier transform. In this case, all the vector fields yield ghost derivatives.

5.1.5. Asymptotically hyperbolic manifolds

We continue to assume that $M$ is a manifold with smooth boundary ${\partial}M$. The groupoid $\mathcal{G}_{ah}$ modeling asymptotically hyperbolic spaces is chosen such that

[TABLE]

Let $\mathcal{L}_{x}:=T_{x}({\partial}M)\rtimes(0,\infty)$ be the semi-direct product as in the previous subsection, with $(0,\infty)$ acting by dilations on $T_{x}({\partial}M)$. Let $\mathcal{L}\to{\partial}M$ be the bundle of Lie groups with fiber $\mathcal{L}_{x}$. Then

[TABLE]

The topology is again locally given by a transformation groupoid. This can be seen in the case of $M=\mathbb{R}^{n-1}\times[0,\infty)$ with the natural action of $G_{n}:=\mathbb{R}^{n-1}\rtimes(0,\infty)$ obtained by recalling that, as smooth manifolds, we have $\mathbb{R}^{n-1}\rtimes(0,\infty)=\mathbb{R}^{n-1}\times(0,\infty)$. This groupoid is a particular case of the edge groupoid, following next, so the reader can consult to the next section for more details. We again have that $\mathcal{G}_{x}=\mathcal{G}_{x}^{x}=\mathcal{L}_{x}$, if $x\in{\partial}M$, and that all derivatives are ghost derivatives at the boundary. The metric is

[TABLE]

where $x$ is the distance to the boundary (close to the boundary) and $h$ is an everywhere smooth metric on $M$, as before.

Corollary 5.4**.**

Let $s\in\mathbb{R}$ and $P\in L_{s}^{m}(\mathcal{G}_{ah};E,F)\supset\Psi^{m}(\mathcal{G}_{ah};E,F)$, with $M$ compact. Denote $M_{x}:=T_{x}{\partial}M\times\mathbb{R}.$ We have

[TABLE]

All geometric differential operators associated to the metric $g_{ah}$ belong to $\operatorname{Diff}^{m}(\mathcal{V}_{ah};E,F)\subset\Psi^{m}(\mathcal{G}_{ah};E,F)$, for suitable vector bundles $E,F\to M$.

We thus need to study the invertibility of certain (right) invariant operators on the group $G_{n-1}:=\mathbb{R}^{n-1}\rtimes(0,\infty)$, for which standard methods of representation theory can be used.

5.2. The edge groupoid

This example is motivated by the results in [32, 34, 54, 69, 92], where the original Fredholm results on the edge calculus can also be found. It is a particular case of the next example, that of a desingularization groupoid, but we nevertheless treat it separately, for the benefit of the reader. See also [22, 43, 104]. We consider the following framework.

First, $M$ is a manifold with smooth boundary ${\partial}M$. We assume that we are given a smooth fibration $\pi:{\partial}M\to B$, where $B$ is a smooth manifold (thus without boundary). We fix a tubular neighborhood $U$ of ${\partial}M$ in $M$: $U\simeq{\partial}M\times[0,1)$. Let $\mathcal{H}:=B\times B$ be the pair groupoid. We now construct the so called edge groupoid $\mathcal{G}_{e}$ that will turn out to be a groupoid with Lie algebroid given by the set $\mathcal{V}_{e}$ of vector fields on $M$ that are tangent to the fibers of $\pi:{\partial}M\to B$ (in particular, these vector fields are tangent to the boundary). If $B={\partial}M$, then we recover the groupoid that models asymptotically hyperbolic spaces. We denote by $\operatorname{Diff}(\mathcal{V}_{e})$ the algebra of differential operators generated by $\mathcal{V}_{e}$ and by multiplication with functions in ${\mathcal{C}}^{\infty}(M)$.

Let $\mathcal{L}:=TB\rtimes\mathbb{R}_{+}^{*}\to B$ be the bundle of Lie groups obtained from $TB\to B$ (regarded as a bundle of commutative Lie groups) by taking the semi-direct product with $\mathbb{R}_{+}^{*}:=(0,\infty)$ acting by dilations on the fibers of $TB\to B$. Its pullback $\pi^{\downarrow\downarrow}(\mathcal{L})$ via $\pi:{\partial}M\to B$ is hence a Lie groupoid with units ${\partial}M$. Let $M_{0}:=M\smallsetminus{\partial}M$ be the interior of $M$. Then, as a set, the edge groupoid $\mathcal{G}_{e}$ is the disjoint union

[TABLE]

To define the smooth structure on this groupoid, we could use either the results in [70] or proceed directly in four steps as follows.

Step 1. We first consider the adiabatic groupoid $\mathcal{H}_{ad}$ of $\mathcal{H}:=B\times B$ (this is the tangent groupoid of [16]; see [69] for more details and references). The adiabatic groupoid $\mathcal{H}_{ad}$ is a Lie groupoid with units $B\times[0,\infty)$ and Lie algebroid $A(\mathcal{H}_{ad})=TB\times[0,\infty)\to B\times[0,\infty)$, which, as a vector bundle, is the fibered pull-back of $A(\mathcal{H})=TB\to B$ to $B\times[0,\infty)$ via the projection $B\times[0,\infty)\to B$. The Lie algebroid structure on the sections of $A(\mathcal{H}_{ad})$ is not that of a fibered pull-back Lie algebroid, but is given by $[X,Y](t)=t[X(t),Y(t)].$ As a set, $\mathcal{H}_{ad}$ is the disjoint union

[TABLE]

The groupoid structure of $\mathcal{H}_{ad}$ is such that $A(\mathcal{H})\times\{0\}$ has the Lie groupoid structure of a bundle of Lie groups and $\mathcal{H}\times(0,\infty)$ has the product Lie groupoid structure with $(0,\infty)$ the groupoid associated to a space (that is $(0,\infty)$ has only units, and all orbits are reduced to a single point). The smooth structure is obtained using the exponential map. See also [102].

Step two. Let $\pi:{\partial}M\to B$ be the given fibration map. We denote also by $\pi$ the resulting map ${\partial}M\times[0,\infty)\to B\times[0,\infty)$. Then we consider the pullback Lie groupoid $\pi^{\downarrow\downarrow}(\mathcal{H}_{ad})$.

Step three. Let $\mathbb{R}_{+}^{*}=(0,\infty)$ act by dilations on the $[0,\infty)$ variable on $\pi^{\downarrow\downarrow}(\mathcal{H}_{ad})$ and consider the semi-direct product $\pi^{\downarrow\downarrow}(\mathcal{H}_{ad})\rtimes\mathbb{R}_{+}^{*}$ [25]. As a set, it is the disjoint union of $\pi^{\downarrow\downarrow}(\mathcal{L})$ and of the pair groupoid of ${\partial}M\times(0,\infty)$.

Step four. Let us identify the tubular neighborhood $U\subset M$ of ${\partial}M$ with ${\partial}M\times[0,1)$. Then we can consider the reduction \mathcal{H}^{\prime}:=\big{(}\pi^{\downarrow\downarrow}(\mathcal{H}_{ad})\rtimes\mathbb{R}_{+}^{*}\big{)}_{U}^{U}. By the previous step, this reduction $\mathcal{H}^{\prime}$ is the disjoint union of $\pi^{\downarrow\downarrow}(\mathcal{L})$ and of the pair groupoid of ${\partial}M\times(0,1)$, which we can view as a subset of $M_{0}\times M_{0}$, the pair groupoid of $M_{0}$. We then glue the reduction \mathcal{H}^{\prime}:=\big{(}\pi^{\downarrow\downarrow}(\mathcal{H}_{ad})\rtimes\mathbb{R}_{+}^{*}\big{)}_{U}^{U} with $M_{0}\times M_{0}$ by identifying the reduction of $\mathcal{H}^{\prime}$ to $U\smallsetminus{\partial}M$ (which is the pair groupoid of ${\partial}M\times(0,1)$, as we have seen), with its image in $M_{0}\times M_{0}$. This glueing construction is, of course, nothing but a particular case of the glueing construction in [35] and [69].

In any case, we obtain right away from the definition that the edge groupoid $\mathcal{G}_{e}$ is a Hausdorff stratified submersion groupoid. The set of units of $\mathcal{G}_{e}$ is $M$ and the representations $\pi_{x}$, $x\in{\partial}M$, are equivalent precisely when they map to the same point in $B$ and they act on $\pi^{-1}(b)\times\mathcal{L}_{b}\simeq\pi^{-1}(b)\times T_{b}B\times\mathbb{R}$.

If $P\in\operatorname{Diff}(\mathcal{V}_{e})$ and $b\in B$, we can obtain the limit operators $P_{b}:=\pi_{b}(P)$ as follows. The restriction of $P$ to an infinitesimal neighborhood of $\pi^{-1}(b)$ in $M$ will have some hidden (ghost) derivatives coming from the Lie algebra of the group $\mathcal{L}_{b}:=T_{b}B\rtimes\mathbb{R}_{+}^{\infty}$. Let us choose local coordinates $(y,z)$ on ${\partial}M$ that are compatible with ${\partial}M\to B$, in the sense that $z$ comes from a coordinate system on $B$. Let $x$ denote the defining function of ${\partial}M$. Then locally, $\mathcal{V}_{e}$ is generated by $x{\partial}_{x}$, ${\partial}_{y_{j}}$, $x{\partial}_{z_{k}}$, with $j$ and $k$ (always) ranging through a suitable index set:

[TABLE]

With this notation, the Lie algebra $\operatorname{Lie}(\mathcal{L}_{b})$ is generated by the vector fields $x{\partial}_{x}$ and $x{\partial}_{z_{k}}$. These come from non-zero vector fields in $\mathcal{V}_{e}$ that restrict to 0 on $\pi^{-1}(b)$, they are the ghost derivatives. We let the ghost derivatives act on $\mathcal{L}_{b}$, and thus we obtain a differential operator $P_{b}$ on $\pi^{-1}(b)\times\mathcal{L}_{b}$. The metric on $M$ is then an edge metric in the sense of Mazzeo [54]. It can be obtained by patching together metrics for which $x{\partial}_{x}$, ${\partial}_{y_{j}}$, $x{\partial}_{z_{k}}$ are an orthonormal set of vectors.

Theorem 4.12 then becomes.

Corollary 5.5**.**

Let $P\in L_{s}^{m}(\mathcal{G}_{e};E,F)\supset\Psi^{m}(\mathcal{G}_{e};E,F)$, $M$ compact. We have

[TABLE]

where $M_{b}:=\pi^{-1}(b)\times T_{b}B\times\mathbb{R}.$ All geometric differential operators associated to $g_{e}$ belong to $\operatorname{Diff}^{m}(\mathcal{V}_{e};E,F)\subset\Psi^{m}(\mathcal{G}_{e};E,F)$, for suitable $E,F\to M$.

If $B$ is reduced to a point and ${\partial}M$ is connected, the groupoid $\mathcal{G}_{e}$ constructed in the last subsection recovers the groupoid of the $b$-calculus: $\mathcal{G}_{e}=\mathcal{G}_{b}$. It models in this case manifolds with cylindrical ends. If $B={\partial}M$, the corresponding groupoid models “asymptotically hyperbolic” spaces.

5.3. Desingularization groupoids

One of the nice features of the class of stratified submersion groupoids is that it is invariant with respect to desingularization along suitable submanifolds. In this subsection, we will give an ad hoc argument for this statement. Recall the fibered pull-back $\pi^{\downarrow\downarrow}(B)$ of Example 2.19. We proceed in a slightly greater generality than in [69], to which we refer for more details and for the unexplained arguments.

Let us assume that $M$ is a manifold with corners, that $H$ is a hyperface of $M$ and that we are given a tame submersion of manifolds with corners $\pi:H\to B$. Let $U\simeq H\times[0,1)$ be a tubular neighborhood of $H$ in $M$. The hyperface $H$ will play the role played by the boundary in the previous examples. Typically, $M$ will be the result of a desingularization procedure, such as a blow-up (see Remark 5.6 and the following section).

We assume that we are given a Lie groupoid $\mathcal{G}$ on $M\smallsetminus H$ and a Lie groupoid $\mathcal{H}\to B$ such that the following is satisfied.

Local fibered pull-back structure assumption: The reduction of $\mathcal{G}$ to $H\times(0,1)$ is isomorphic to the pullback $p^{\downarrow\downarrow}(\mathcal{H})$ via the map $p:=\pi\circ p_{1}:H\times(0,1)\to B$.

We fix $\mathcal{H}$ as in the above assumption throughout this subsection. We define then a groupoid $\mathcal{K}\rightrightarrows M$ such that, as a set, it is the disjoint union

[TABLE]

The groupoid $\mathcal{H}$ replaces the pair groupoid $B\times B$ of the example of the edge groupoid. We proceed as in that example to consider the adiabatic groupoid $\mathcal{H}_{ad}$ of $\mathcal{H}$, which has units $B\times[0,\infty)$. We pull back this groupoid to a groupoid $\pi^{\downarrow\downarrow}(\mathcal{H}_{ad})$ using the map $\pi_{1}:H\times[0,\infty)\to B\times[0,\infty)$, and then we consider the semi-direct product $\pi_{1}^{\downarrow\downarrow}(\mathcal{H}_{ad})\rtimes\mathbb{R}^{*}_{+}=\pi_{1}^{\downarrow\downarrow}(\mathcal{H}_{ad}\rtimes\mathbb{R}^{*}_{+})$ with $\mathbb{R}_{+}^{*}$ acting by dilations on $[0,\infty)$.

As in the previous example of the edge groupoid, we view $U$ as an open subset of $H\times[0,\infty)$ and we consider the groupoid $\mathcal{H}^{\prime}$ defined as the reduction to $U$ of $\pi^{\downarrow\downarrow}(\mathcal{H}_{ad})\rtimes\mathbb{R}^{*}_{+}$, with $\mathcal{H}$ as in the local fibered pull-back assumption. The construction is completed as in the fourth step of the edge groupoid. That is, we use the invariant subset $H\times\{0\}$ of the units of $\mathcal{H}^{\prime}$ to write the groupoid as a disjoint union using Remark 2.7. The reduction of $\mathcal{H}^{\prime}$ to the complement of $H\times\{0\}$, that is, to $H\times(0,1)$, is (isomorphic to) the fibered pull-back $p^{\downarrow\downarrow}(\mathcal{H})$ of $\mathcal{H}$ to $H\times(0,1)$, by the local fibered pull-back structure assumption. We can view this fibered pull-back $p^{\downarrow\downarrow}(\mathcal{H})$ as a subset of $\mathcal{G}$. This gives that we can glue $\mathcal{H}^{\prime}$ and $\mathcal{G}$ along the common open groupoid $p^{\downarrow\downarrow}(\mathcal{H})$ to obtain a groupoid $\mathcal{K}$.

It can be proved that if $\mathcal{G}$ is Fredholm, then $\mathcal{K}$ is also Fredholm. Moreover, if $\mathcal{G}$ is a stratified submersion groupoid, then $\mathcal{K}$ will also be a stratified submersion groupoid.

Remark 5.6*.*

Typically, we start with a Lie groupoid $\mathcal{G}^{\prime}\rightrightarrows M^{\prime}$ and $L\subset M^{\prime}$ is a submanifold with corners. Then $M:=[M^{\prime}:L]$, the blow-up of $M^{\prime}$ with respect to $L$ and $H$ is the hyperface corresponding to $L$ in this blow-up, with $\pi:H\to L$ the blow-down map. Finally, $\mathcal{G}$ is the reduction of $\mathcal{G}^{\prime}$ to the complement of $L$. Then we denote $[[\mathcal{G}^{\prime}:L]]:=\mathcal{K}$. This is the desingularization construction from [69].

5.4. Desingularization and singular spaces

Let us show how to use the desingularization construction in some typical examples. By $M_{k}$ and $L_{k}$ we will denote manifolds with corners of depths $k$. Thus $M_{0}$ and $L_{0}$ will have, in fact, no corners or boundary (hence they will be “smooth”).

5.4.1. The desingularization of a smooth submanifold

The simplest example of a desingularization groupoid is the desingularization of the pair groupoid $\mathcal{G}_{0}:=M_{0}\times M_{0}$ with respect to a smooth submanifold $L_{0}\subset M_{0}$. Recall that $M_{0}$ is also smooth. Thus neither $M_{0}$ nor $L_{0}$ have corners. The example of this subsection is a particular case of the edge groupoid, and hence it is related to the edge calculus. [34, 54, 66, 92].

Let $N$ be the normal bundle of $L_{0}$ in $M_{0}$ and denote by $S\subset N$ the set of unit vectors in $N$, that is, $S$ is the unit sphere bundle of the normal bundle of $L$ in $M_{0}$. We let $\pi:S\to L_{0}$ be the natural projection. Then the blow-up $M_{1}:=[M_{0}:L_{0}]$ of $M_{0}$ with respect to $L_{0}$ is the disjoint union

[TABLE]

with the structure of a manifold with smooth boundary $S$. We let $\mathcal{G}_{1}$ to be the associated edge groupoid introduced in Subsection 5.2. It is a Lie groupoid with base $M_{1}:=[M_{0}:L_{0}]$. We are moreover in the framework of Remark 5.6, so $\mathcal{G}_{1}=[[\mathcal{G}_{0}:L_{0}]]$. The filtration of $M_{1}$ has two sets, namely, $U_{1}=M_{1}$ and $U_{0}:=M_{0}\smallsetminus L_{0}\subset M_{1}$, both of which are open and invariant for $\mathcal{G}_{1}$ (but $U_{0}$ is not invariant for $\mathcal{G}_{0}$, in general).

It turns out that $\mathcal{G}_{1}$ is a stratified submersion Lie groupoid if we consider the two strata $U_{0}$ and $S$. Also, $(\mathcal{G}_{0})_{U_{0}}^{U_{0}}=U_{0}\times U_{0}$ and $\mathcal{G}_{S}$ is the fibered pull-back of a bundle of Lie groups $\mathcal{L}\to S$, as described in Subsection 5.2.

5.4.2. The desingularization of a submanifold with boundary

We now generalize he last construction to manifolds with boundary. Let $M_{1}$ be a compact manifold with smooth boundary. We denote by $F:={\partial}M_{1}$ its boundary and by $G:=M_{1}\smallsetminus F$ its interior. On $M_{1}$ we consider the Lie algebra of vector fields $\mathcal{V}_{b}$ tangent to $\partial M_{1}$, as before. It is the space of sections of $T^{b}M_{1}$, the “$b$-tangent bundle” [57, 59] of $M_{1}$. Let $\mathcal{G}_{1}:=\mathcal{G}_{b}$, as defined in the previous section.

Let $L_{1}\subset M_{1}$ be an embedded smooth submanifold assumed to be such that its boundary is ${\partial}L_{1}=L_{1}\cap{\partial}M_{1}$ and such that $L_{1}$ intersects ${\partial}M_{1}$ transversely. Then $L_{1}$ has a tubular neighborhood $U$ in $M_{1}$, and hence we can consider the blow-up $M_{2}:=[M_{1}:L_{1}]$, which will be a manifold with corners of codimension 2. Moreover, the reduction of $\mathcal{G}_{1}=\mathcal{G}_{b}$ to $U\smallsetminus L_{1}$ satisfies the local fibered pull-back structure assumption, since it is the $b$-groupoid of $U\smallsetminus L_{1}$. In view of Remark 5.6, we can then define $\mathcal{G}_{2}:=[[\mathcal{G}_{1}:L_{1}]]$.

Again it turns out that $\mathcal{G}_{2}$ is a stratified submersion Lie manifold; indeed, this is seen by choosing the following filtration of $M_{2}$ with three sets:

[TABLE]

The sets $U_{j}$ are open and $\mathcal{G}_{2}$ invariant (but not $\mathcal{G}_{1}$ invariant). The sets $U_{0}$ and $U_{1}$ are $\mathcal{G}_{2}$-invariant by construction. Assume they are connected, for simplicity. The restriction $(\mathcal{G}_{2})_{U_{1}}$ coincides with the reduction $(\mathcal{G}_{1})_{U_{1}}^{U_{1}}$ by the definition of the desingularization groupoid (Remark 5.6). In particular, $(\mathcal{G}_{2})_{U_{0}}=(\mathcal{G}_{2})_{U_{0}}^{U_{0}}=(\mathcal{G}_{1})_{U_{0}}^{U_{0}}=U_{0}^{2}$ and

[TABLE]

where $U_{0}^{2}$ and $(U_{1}\smallsetminus U_{0})^{2}$ are pair groupoids. In the general case, if $U_{1}\smallsetminus U_{0}$ is not connected, we write $U_{1}\smallsetminus U_{0}=\sqcup V_{j}$ as the disjoint union decomposition of its connected components, then we have $(\mathcal{G}_{2})_{U_{1}\smallsetminus U_{0}}\,:=\,\sqcup_{j}V_{j}^{2}\times\mathbb{R}_{+}^{*}$. Let us denote as in the boundaryless case by $S$ the unit sphere bundle of the normal bundle to $L_{1}$ in $M_{1}$. (The set $S$ is the last stratum $U_{2}\smallsetminus U_{1}$ in $M_{2}$.) Then the restriction of $\mathcal{G}_{1}$ to $S:=M_{2}\smallsetminus U_{1}$ is the following fibered fibered pull-back groupoid. Let $\pi:S\to L$ be the natural projection, as before. Let again $G_{S}:=T^{b}L\rtimes\mathbb{R}_{+}^{*}\to L$ be the group bundle over $L$ obtained by taking the direct product of the $b$-tangent bundle to $L$ with the action of $\mathbb{R}_{+}^{*}$ by dilation on the fibers of $TL\to L$. Then

[TABLE]

5.4.3. Desingularization of a stratified subset of dimension one

We now deal with a slightly more complicated example by combining the two previous examples. We thus introduce the groupoid that is obtained from the desingularization of a stratified subset of dimension one. Full details as well as applications will be included in a forthcoming paper with Mihai Putinar.

Let $M_{0}$ be a smooth, compact manifold (so no corners). Let $L_{0}:=\{P_{1},P_{2},\ldots,P_{k}\}\subset M_{0}$ and let us assume that we are given a subset $\mathcal{S}\subset M_{0}$ such that

[TABLE]

where each $\gamma_{j}$ is the image of a smooth map $c_{j}:[0,1]\to M_{0}$, with the following properties:

1. (i)

$c_{j}^{\prime}(t)\neq 0$, 2. (ii)

$c_{j}(0),c_{j}(1)\in L_{0}:=\{P_{1},P_{2},\ldots,P_{k}\}$, 3. (iii)

$c_{j}((0,1))$ are disjoint and do not intersect $L_{0}$ and 4. (iv)

the vectors $c_{j}^{\prime}(0)$ and $c_{j}^{\prime}(1)$, $j=1,\ldots,l$, are all distinct.

We now introduce the desingularization of $\mathcal{G}_{0}:=M_{0}\times M_{0}$ (the pair groupoid) with respect to $\mathcal{S}$. The set $L_{0}:=\{P_{1},P_{2},\ldots,P_{k}\}\subset M_{0}$ satisfies the assumptions of 5.4.1. We can first define $\mathcal{G}_{1}$ to be the desingularization of $\mathcal{G}_{0}$ with respect to $L_{0}$ as in that example:

[TABLE]

which is a groupoid with units $M_{1}:=[M_{0}:L_{0}]$. The smooth maps $c_{j}$ then lift to smooth maps

[TABLE]

The assumption that the vectors $c_{j}^{\prime}(0)$ and $c_{j}^{\prime}(1)$, $j=1,\ldots,l$, are all distinct then gives that the sets $\tilde{\gamma}_{j}:=\tilde{c}_{j}([0,1])$ are all disjoint and intersect the boundary of $M_{1}$ transversally. Let $L_{1}$ be the disjoint union of the embedded curves $\tilde{\gamma}_{j}$. Then we can perform a further desingularization along $L_{1}$, as in 5.4.2, thus obtaining

[TABLE]

which is a boundary fibration Lie groupoid. The Lie groupoid $\mathcal{G}_{2}$ is the desingularization groupoid of $M_{0}$ with respect to $\mathcal{S}$. Its structure is given as in the previous subsection.

This example can be extended to higher dimensional cases by using clean intersecting families.

Acknowledgements

We thank Vladimir Georgescu, Marius Mantoiu, and Wolfgang Schulze for useful discussions.