Fredholm criteria for pseudodifferential operators and induced representations of groupoid algebras

We characterize the groupoids for which an operator is Fredholm if, and only if, its principal symbol and all its boundary restrictions are invertible. A groupoid with this property is called {\em Fredholm}. Using results on the Effros-Hahn conjecture, we show that an almost amenable, Hausdorff, second countable groupoid is Fredholm. Many groupoids, and hence many pseudodifferential operators appearing in practice, fit into this framework. In particular, one can use these results to characterize the Fredholm operators on manifolds with cylindrical and poly-cylindrical ends, on manifolds that are asymptotically Euclidean or asymptotically hyperbolic, on products of such manifolds, and on many others. Moreover, we show that the desingularization of Lie groupoids preserves the class of Fredholm groupoids.


Introduction
We obtain necessary and sufficient conditions for operators modeled by groupoids to be Fredholm. Examples include operators obtained by desingularization of singular spaces by successively blowing up the lowest dimensional singular strata. We begin with a general study of Fredholm conditions for pseudodifferential operators in the framework of Fredholm groupoids. A Fredholm groupoid is, by definition, a locally compact groupoid with a Haar system for which the Fredholm property is equivalent to the invertibility of the principal symbol and of its fiberwise boundary restrictions. We obtain a general characterization of Fredholm groupoids. In particular, using some results of Renault [31,32] and Ionescu and Williams [13], we show that an almost amenable, second-countable, Hausdorff groupoid is Fredholm.
Let G be a groupoid with base M modeling the analysis on some singular space. An A(G)-tame submanifold L ⊂ M is one that has, by definition, a tubular neighborhood on which A(G( becomes a pull-back Lie algebroid. The "desingularization" [[G : L]] of G along L [24] is the a groupoid model-ling the analysis on the space obtained by blowing-up L. The space of units of the desingularization [[G : L]] is [M : L], the blow-up of M along L. The desingularization groupoid is not a blown-up space, however. We use the explicit structure of the desingularized groupoid [[G : L]] (see [24]) to show that it is Fredholm if G is. Our results specialize to yield Fredholm conditions for operators on manifolds with cylindrical and poly-cylindrical ends, on manifolds that are asymptotically Euclidean or asymptotically hyperbolic, on products of such manifolds, on manifolds that locally at infinity are products of such manifolds, and on others. Most of the (generally) easy proofs are contained in [25], this paper being a summary and update of some results in that paper. We thank Ingrid and Daniel Beltiţȃ

Fredholm groupoids
Recall that a groupoid G is a small category in which every morphism is invertible. We shall write G ⇒ M for a groupoid with objects (or units) M . The domain and range of a morphism therefore give rise to maps d, r : G → M . We refer to [21,30] for the results and concepts used-but not recalled-in this paper. Let G ⇒ M be a locally compact groupoid endowed with a Haar system (λ x ) x∈M . We denote by C * (G) the C * -algebra of G and by C * r (G) the reduced C * -algebra of G. Also, we denote G A := d −1 (A) and As usual, we associate to any x ∈ M the regular representation π x : All the morphisms (and representations) in this paper will preserve the involution.
Example 2.1. Recall that the pair groupoid H := A × A is the groupoid having exactly one arrow between any two units. Let G ⇒ L be a groupoid and f : M → L. An important generalization of the pair groupoid is the fibered pull-back groupoid: Both M 0 and F are uniquely determined by G, so this notation will remain fixed in what follows. Also, in Definition 2.2, all representations π x0 , x 0 ∈ M 0 , are unitarily equivalent to the vector representation π 0 : C * (G) → L(L 2 (M 0 )) obtained by identifying r : G x0 ≃ M 0 .
Recall [33] that if A is a C * -algebra with unit, then a set F of representations of A is called invertibility sufficient if the following condition is satisfied: "a ∈ A is invertible if, and only if, φ(a) is invertible for all φ ∈ F ." If A does not have a unit, we replace A with A + := A ⊕ C and F with F + := F ∪ {χ 0 : A + → C} [26]. The following two results give a first characterization of Fredholm groupoids.
For index theory, the first two conditions of the theorem are enough. Invertibility sufficient families of representations consist of non-degenerate representations. A non-degenerate representation of a (closed, two-sided) ideal in a C * -algebra has a unique extension to the whole algebra.
The following strong converse of Theorem 2.3 holds true.
Theorem 2.4. Let G ⇒ M be a locally compact groupoid satisfying the three conditions (i-iii) of Theorem 2.3. Then, for any unital C * -algebra Ψ containing C * r (G) as an essential ideal and for any a ∈ Ψ, we have that π 0 (a) if Fredholm if, and only if, π x (a) is invertible for each x / ∈ M 0 and the image of a in Ψ/C * r (G) is invertible.

Relation to the Effros-Hahn conjecture
We now want to obtain some more concrete and easier to use conditions for a groupoid G to be Fredholm. We shall say that a locally compact groupoid G has the weak inclusion property (wi-property, for short) if every irreducible representation of C * (G) is weakly contained in a representation of C * (G) induced from a representation of an isotropy subgroup G y y := d −1 (y) ∩ r −1 (y) [13,32] (equivalently, if every primitive ideal of C * (G) contains an ideal induced from a representation of an isotropy subgroup G y y ). A groupoid G with the wi-property and such that all the groups G y y are amenable will be called EHamenable. Recall that a locally compact groupoid G satisfies the generalized Effros-Hahn (EH) conjecture if every primitive ideal of C * (G) is induced from a representation of an isotropy group G y y [8,13,14,30,40]. Example 3.1. Let G ⇒ B be a locally trivial bundle of groups (so d = r) with typical fiber a locally compact group G. Also, let f : M → B be a continuous map that is a local fibration. Then f ↓↓ (G) is a locally compact groupoid with a Haar system that satisfies the generalized EH conjecture, and hence it has the wi-property. It will be EH-amenable if, and only if, the group G is amenable.
Recall that a groupoid is called metrically amenable if the canonical surjection C * (G) → C * r (G) is injective [38]. We shall need two results from [26] (see also [9]). Proposition 3.2. Let G ⇒ F be an EH-amenable locally compact groupoid. Then the family of regular representations {π y , y ∈ F } of C * (G) is invertibility sufficient. In particular, G is metrically amenable.
The class of EH-amenable groupoids is closed under extensions. The same holds if one replaces "is EH-amenable" with "satisfies the generalized EH conjecture" or "has the wi-property." The following result leads to more applicable Fredholm conditions.
Ui is topologically amenable for all i, then we shall say that the locally compact groupoid G ⇒ M is almost amenable. By combining the above two propositions with the proof of the generalized EH conjecture [13,31,32] for amenable, Hausdorff, second countable groupoids, we obtain the following result. We are interested in Fredholm groupoids because of their applications to Fredholm conditions. Let G be a continuous family groupoid [15] and Ψ m (G) be the space of order m, classical pseudodifferential operators P = (P x ) x∈M on G [15] (see [2,1,22,27,39] for Lie groupoids, which are continuous family groupoids). Recall that, by definition, each P x ∈ Ψ m (G x ), x ∈ M . Also, P acts on M 0 via P x0 : The following result is interesting only in the case M compact.
This theorem is proved by considering a := (1 + ∆) (s−m)/2 P (1 + ∆) −s/2 , which belongs to the closure of Ψ 0 (G), by the results in [17]. For the next theorem, however, one has to consider the Cayley transform of P instead of the operator a. Theorem 3.7. Let G ⇒ M be as in Theorem 3.6 and let P ∈ Ψ m (G) be an elliptic operator. Then its essential spectrum is The above two theorems extend to operators acting on vector bundles on M . The operators P x are the analogues in our setting of the "limit operators" considered in [5,28] and many other references. See also [3,6,10,11,15,16,17,18,19,20,23,29,34,35,36,37] and the references therein for related results.

Desingularization and Fredholm conditions
We want an ample supply of Fredholm groupoids. In this section, we recall the desingularization procedure along a "tame" submanifold of the set of units of a Lie groupoid [24]. Recall the thick pullback π ↓↓ (B) of Example 2.1.
Definition 4.1. Let A → M be a Lie algebroid. An A-tame submanifold L ⊂ M is a submanifold that has a tubular neighborhood L ⊂ U , such that there exists a Lie algebroid B → L and an isomorphism A| U ≃ π ↓↓ (B) over U , π : U → L.
By A(G) we denote the Lie algebroid of a Lie groupoid G. We have the following structure result in a neighborhood of a tame submanifold. . The groupoid structure of H ad is such that A(H) × {0} has the Lie groupoid structure of a bundle of Lie groups and G × (0, ∞) has the product Lie groupoid structure (that is (0, ∞) has only units, and all orbits are reduced to a single point).
Step three. Let R * + = (0, ∞) act by dilations on the [0, ∞) variable on π ↓↓ (H ad ) and consider the semi-direct product π ↓↓ (H ad ) ⋊ R * + [7]. We then define (1) [  W is isomorphic to the fibered pull-back π ↓↓ (A(K) ⋊ R * + ) to S via the natural projection π : S → L, where A(K) ⋊ R * + is regarded as a bundle of Lie groups. The inclusion G W W → G induces an isomorphism C * (G W W ) ≃ C * (G). One sees that the resulting glued set is a Hausdorff groupoid as in [12].  This theorem can be used to obtain Fredholm conditions for operators on polyhedral domains, as well as on some other stratified spaces.