Singularity structures for noncommutative spaces

We introduce a (bi)category $\mathfrak{Sing}$ whose objects can be functorially assigned spaces of distributions and generalized functions. In addition, these spaces of distributions and generalized functions possess intrinsic notions of regularity and singularity analogous to usual Schwartz distributions on manifolds. The objects in this category can be obtained from smooth manifolds, noncommutative spaces, or Lie groupoids. An application of these structures relates the longitudinal propagation of singularities for pseudo-differential operators on a groupoid with propagation of singularities on the base manifold.


Introduction
A precise description of propagation of singularities under the solution operators of hyperbolic partial differential equations depends on the formulation of the notions of singular support and wavefront sets for distributions. The propagation of singularities can then be viewed as a link between the wave and corpuscular theory of light. There are many applications in geometry and analysis of this phenomenon particularly to spectral theory of elliptic pseudo-differential operators. These phenomena are even more remarkable in the case of manifolds with boundaries or corners (see [15,16,23]). In principle, it is evident that propagation of singularities occurs as the solution operators to hyperbolic-partial differential equations are constructed via a parametrix construction related to some calculus of pseudo-differential operators, whereby the dynamics is transferred to the space of principal-symbols. In many situations involving manifolds with corners the relevant calculus can be constructed based on differentiable groupoids [19,20,14]. In this paper we shall present a category theoretic view of distribution theory including the singularities and propagations of singularities . We shall also consider morphisms that relate these propagations. As an application we shall related the longitudinal propagation on certain groupoids to the transverse propagation on the base manifold with respect to vector representation.
First we consider a category Sing whose objects, referred to as singularity structures, are triples (A, X, Y ) where X and Y are suitable Fréchet modules over a filtered algebra A. In this category an instance of a closed manifold M is represented by A = Ψ ∞ (M ), the algebra of (classical) pseudodifferential operators, X = Ψ −∞ (M ), the ideal of regularizing operators and Y = C ∞ (M ). More generally, objects in Sing are provided by noncommutative spaces represented by (regular) spectral triples in the sense of Connes, geometric Hilbert spaces and groupoids.
The association of distributions to a singularity structure follows from a simple observation. In the classical case on M represented as above, the space of distributions, that is the dual space to the space of densities D ′ (M ) = |Λ|(M ) ′ can be realized exactly as a left A module map between X and Y . As a direct generalization we consider abstract generalized functions and distributions on any singularity structure (A, X, Y ) and obtain in a natural way the notion of regularity and singularity of abstract distributions and generalized functions associated to a general triple (A, X, Y ). Thus Sobolev type regularity (by choice of a hypo-elliptic, or a self-adjoint positive operator and a scale) and constructions of singular supports and wavefront sets are built in by definition. The Supported by FWF grant P20525 of the Austrian Science Fund. Supported by FWF grants Y237-N13 and P20525 of the Austrian Science Fund. regularity can be described entirely by the graded Fréchet space structure on X and Y , however the singularities depend crucially on the choice of the algebra A.
In the classical analysis of linear PDEs the main benefit of singularity and regularity analysis of distributions lies in the use of their compatibility with respect to pull-backs and push-forwards of distributions under appropriate smooth maps.(See [8] Chapter 6 for some interesting applications.) Thus a class of morphisms is introduced so that the microlocal information defined for an (A, X, Y ) generalized function behaves functorially with respect to them. The obvious class of morphisms of these singularity structures is very restrictive, hence it is necessary to have a larger class of morphisms essentially in the spirit of Morita equivalence. It is expected that suitably regular KK-cycles, possibly with additional structure, will provide morphisms of the singularity structures, especially the ones arising from spectral triples. Here we shall focus on morphisms defined out of vector-representations on Lie-groupoids. This is the most relevant case to understand remarkable propagation results in, e.g., [11,15,23]. However the assumptions in this article are much stronger, which simplifies the exposition. We expect similar results to hold for differential groupoids associated to manifolds with corners.
The interpretation of the propagation of singularity phenomenon in this new context is akin to an action by R. This can be visualized as an abstract Egorov theorem which in the classical context provides an R action on the singularity structure associated to a manifold as above. Thus comparison of the dynamics associated to various singularity structures with R action is now possible via considering equivariant morphisms between them. As mentioned already an application of the category Sing is to show that the anchor map on the Lie algebroid A(G) relates the longitudinal propagation on a groupoid with the transverse propagation on the base manifold.

Algebraic basics
This section introduces the notation used in this article. We shall also collect some examples for easy reference.
We shall consider an algebra A, a right A-module X and a left A-module Y given to us. The space of (set-theoretic) maps from X to Y will be denoted by M(X, Y ) and is given a natural A-bimodule structure. For any φ : X → Y , and a ∈ A this structure is defined by For our purposes both the modules X and Y shall be Fréchet spaces provided with a grading. Recall that a grading on a Fréchet space X is a sequence of seminorms n that is increasing 1 ≤ 2 ≤ . . . and such that it generates the locally convex topology on X (cf. [9], Def. 1.1.1). Additionally we shall assume that the algebra A is filtered. This entails that there exist subspaces Definition 2.1. A (right) module X over a graded algebra A is called a Fréchet module if X is a graded Fréchet space and for each P ∈ A j there exists a constant C > 0 such that for all n ∈ N: Analogous notion for left and bi-modules over A will be understood.
The following two examples provide a strong motivation for the construction developed in this paper.
Example 2.2. Let A = Ψ ∞ (M ) be the algebra of (classical) pseudo-differential operators on a closed manifold M with the usual filtration given by the degree of the operator.
We shall grade C ∞ (M ) with Sobolev norms. To this end, let ∆ denote the Laplace operator on M associated to some Riemannian metric on M and set With the above grading C ∞ (M ) is a graded Fréchet algebra in the sense of Definition 3.3 below.
For any operator T : L 2 (M ) → L 2 (M ) let D HS denote its Hilbert-Schmidt norm. Given integers p, q we set T p,q := (1 + ∆) Throughout this article the grading on Ψ −∞ (M ) will be defined by: Since Ψ −∞ (M ) is a two sided ideal in A = Ψ ∞ (M ), we shall take this module structure and consider the natural (left) action of A on C ∞ (M ). Both C ∞ (M ) and Ψ −∞ (M ) are then Fréchet modules over A We note that by the Schwartz kernel theorem, given a choice of non-vanishing density on M one can identify Ψ −∞ (M ) with C ∞ (M × M ). The above grading on Ψ −∞ (M ) is then equivalent to the tensor-product grading with the chosen Sobolev grading on C ∞ (M ).
Next we consider a more general example coming from a regular spectral triple in the sense of Connes [4]. Let D be an unbounded self-adjoint operator on a Hilbert space H with compact resolvent. Let ∆ := D 2 . We can define a Sobolev space H s for s ∈ R as usual by means of completion with respect to the seminorms The core of D, given by H ∞ := s H s is then naturally a Fréchet space.
3. An algebra A of pseudodifferential operators such that H ∞ is a Fréchet A module can be obtained from an involutive algebra A of bounded operators on H which satisfies a certain regularity condition, namely that (A, H, D) form a regular spectral triple. We briefly recall the notion of a regular spectral triple and the associated algebra of pseudo-differential operators. To describe this algebra of pseudo-differential operators we follow the presentation in [10].
A spectral triple consists of an involutive algebra of operators A on a Hilbert space H and a specified unbounded densely defined self-adjoint operator D such that D has compact resolvent, the elements in We begin by defining an algebra of differential operators D := ∞ k=0 D k (A) for a regular spectral triple. First set the algebra of order zero differential operators to be D 0 := A+[D, A] and inductively define a filtered algebra D := k D k by Definition 2.4. We shall call an operator T on H ∞ a basic pseudo-differential operator of order k if for any l ∈ Z there exist m ∈ Z, R and X so that where X ∈ D and order(X) ≤ k − m and the operator R : H s → H s−l is bounded for all s ∈ R.
More generally, a pseudo-differential operator is a finite linear combination of basic pseudodifferential operators. The algebra of all pseudo-differential operators shall be denoted by A := A (A,H,D) . 3 The following result is a source of examples of Fréchet modules. Proposition 2.5. Let (A, H, D) be a regular spectral triple such that A maps H ∞ to itself. Then H ∞ is a Fréchet module over A.
Proof. An equivalent statement is proved in [6] Appendix B. We shall follow the presentation in [10] and give a proof here for completeness.
Since the spectral triple is regular we can define an algebra Ψ generated by all elements of the form . By definition all elements of Ψ extend to bounded operators on H and δ(Ψ) ⊆ Ψ. Then we can define a new algebra B of elements of the form b|D| k where b ∈ Ψ and k ≥ 0. Note that which proves that B is a filtered algebra with B k generated by elements of the form b|D| j with j ≤ k It is clear that H ∞ is a graded Fréchet module over B. This holds as for any v ∈ H ∞ and b|D| j ∈ B k for j ≤ k we have, where the last inequality follows from spectral theory.
We now claim that the algebra A is a filtered subalgebra of B. To this end it suffices to show that the algebra of differential operators D is a filtered subalgebra of B. We first note that D 0 ⊆ B 0 . Based on this we may use induction to see that if D k−1 ⊆ B k−1 , the calculation Since the operator D is assumed to have compact resolvents, it follows that H ∞ is a Montel space. The consequence of interest is that H ∞ is reflexive. Thus we consider the space of linear maps from the dual space H −∞ = s H s to H ∞ as smoothing operators. In addition when the spectral triple is p-summable then H ∞ is nuclear. This justifies that we define 1 Here we take the tensor product graded Fréchet structure on Ψ −∞ (A, H, D). The essential idea of the proof is the same as in the special case in Example 3.8 below.

Abstract Regularity
We first consider regularity of maps between Fréchet spaces. We refer to [9] for further study.
Definition 3.1. Let X and Y be graded Fréchet spaces. We denote by . n and . ′ n the nth graded norm on X and Y , respectively. We say that a Fréchet smooth map φ : X → Y is polynomially tame if there exist b, k ∈ N and some r ∈ Z such that Here C n > 0 is a constant that depends only on n. If k = 1 we call φ linearly tame. The number r is called the degree of tameness and the set of all maps of tameness degree r is denoted by PT r (X, Y ). Then clearly PT r (X, Y ) ⊆ PT s (X, Y ) for r ≤ s. We set PT(X, Y ) := r PT r (X, Y ). 1 We denote here by⊗ the projective tensor product Remark 3.2. Note that a nontrivial linear map φ is polynomially tame if and only it is linearly tame. Indeed, if k was greater 1 in (2) then replacing x by λx (λ ∈ R) would entail φ = 0.
A polynomially tame map is called regular if it is tame of all orders. We denote by Reg(X, Y ) all regular maps between X and Y , i.e., We will also require the following tameness property for an associative multiplication on a Fréchet space. Definition 3.3. We say that X is a Fréchet algebra if it is a Fréchet space with an associative product and the multiplication is jointly continuous. A graded Fréchet algebra is a Fréchet algebra that is a graded Fréchet space and such that the multiplication satisfies the following tameness condition: there exist b, r 1 , r 2 ∈ N such that With the above definition the following lemma is self-evident.
Lemma 3.4. Let Y be a graded Fréchet algebra and let X be any graded Fréchet space. Then the space PT(X, Y ) of polynomially tame maps from X to Y is an algebra under pointwise operations.
The algebra of polynomially tame maps PT(X, Y ) and the regular maps Reg(X, Y ) depend on not just the topology of X and Y but also on the choice of the grading structures on them. Equivalent gradings (i.e. those with identity map linearly tame of tameness degree 0) provide the same algebras.
The above notions are motivated by the following simple examples: Proof. Let u ∈ H k (M ) be a distribution. Then: whenever n ≥ max(k/2, 2k) =: b. Thus the tameness estimate is satisfied with this b and r = −k.
An important further extension of this result is the following characterization of regular distributions: Theorem 3.7. The only distributions which give rise to regular maps are smooth functions: Proof. In view of Prop. 3.6 C ∞ (M ) ⊆ Reg(M ). Thus it remains to show that given a nonsmooth distribution u ∈ D ′ (M ) the map Θ u is not contained in Reg(M ).
We first observe that there exists some s ∈ R such that v = (1 + ∆) s u ∈ L 2 (M ) and check that if Θ u ∈ Reg(M ) then so is Θ v . Thus without loss of generality we may assume that u ∈ L 2 (M ). Let {φ i } be an orthonormal basis of eigenvectors of ∆ (with eigenvalues λ i ≤ λ i+1 nondecreasing). Then u = n a n φ n (a n ) ∈ ℓ 2 (a n ) ∈ S (N).
We consider the action of Θ u on smoothing operators which are represented by a diagonal matrix with respect to the basis Suppose now that Θ u ∈ Reg(M ). By Remark 3.2 this means that for any degree r of regularity there exist b r in N such that for each k ≥ b r + |r| there exists some C > 0 such that for each (k n ) ∈ S (N) we have Noting that by Weyl's estimates the λ n asymptotically grow polynomially we may rescale (k n ) by (1 + λ n ) k/2 in this estimate to obtain: Inserting k n = δ mn this implies (a n ) ∈ S (N), contradicting our assumption.
Example 3.8. Let P be a pseudodifferential operator of order m. We first observe that right multiplication by P on Ψ −∞ (M ), i.e., the map T → T P is polynomially tame of tameness degree m. To see this one notes that the operator (1 + ∆) m 2 generates Ψ m (M ) as a left-module (and also as a right-module) over Ψ 0 (M ). Therefore we may write By the same token, for k an integer multiple of 1 2 we find an order 0 operator T k such that Putting this together we have In particular this implies that T P n ≤ C T n+m .
The following is a regularity result similar to Th. 3.7: Proposition 3.9. With the notations introduced above, ). Then by Lemma 3.4 given any distribution u ∈ D ′ (M ) it follows that Θ u • P = Θ P u ∈ Reg(M ). By Th. 3.7 this implies that P u ∈ C ∞ (M ) for all distributions u and hence P ∈ Ψ −∞ (M ).
Thus we see that regularity on maps between graded Fréchet spaces gives back usual notions of regularity in familiar cases.
3.1. Singularity structures. As we have seen above, to introduce a notion of regularity one only needs graded Fréchet spaces and a notion of tameness of maps between them. But to further associate "singularities" we need, in addition, a Fréchet module structure with respect to a filtered algebra A. These singularities are defined as ideals in the algebra B = A 0 /A −1 . To this end let σ : A 0 → B denote the symbol homomorphism.
The following result is immediate from the definitions: Proposition 3.10. Let X be a right Fréchet A module and Y a left Fréchet A module. Then for each element a ∈ A j the right module action on P T (X, Y ) has the following effect on the regularity: Definition 3.11. Let X be a right Fréchet A module and Y a left Fréchet A module. We shall refer to the data (A, X, Y ) as a singularity structure.
In the sequel we shall drop the word Fréchet for the sake of readability but all modules will be assumed to satisfy Definition 2.1.
In case B is the commutative algebra C ∞ (X) over some smooth manifold X, we shall associate to any subset Y ⊆ X the ideal of all functions vanishing on Y . Thus for instance the classical wave-front set of a distribution on a manifold M will be represented by an ideal in C ∞ (S * M ) as we describe below.
We shall see below that this notion reproduces the usual notion of wavefront set and even singular support for distributions and provides a "propagation of singularities" result in the context of Weyl algebras, which agrees with usual propagation of singularities for distributions under appropriate conditions.
The remainder of this section provides the motivation for the above definitions by examining classical examples of singularity structures.
3.1.1. Singular support of a distribution.
and only if f vanishes on singsupp(u), and by convention we identify a closed subset with the ideal of all smooth functions that vanish on it, the result follows.

3.2.
Wave-Front set of a distribution. Let us consider the singularity structure defined by the triple A = Ψ ∞ (M ), the algebra of classical pseudo-differential operators on a manifold M and (as in the last example) X = Ψ −∞ (M ) and Y = C ∞ (M ) with the appropriate natural right and left module structure from A. On A we use the usual filtration by the order of the operators. The symbol homomorphism then indeed assigns its symbol to any order 0 pseudodifferential operator. We are going to show that WF A in this case is the usual wavefront set of a distribution.
Given a pseudodifferential operator P we write σ(P ) for the principal symbol of P . Moreover, we denote as usual the characteristic set of P by Char(P ) = σ(P ) −1 ({0}) ⊆ T * M .
Proof. Since Θ u · P = Θ P u it follows from Th. 3.7 that  These are precisely the directions in which P is microlocally elliptic.
3.4. Spectral triples. One may naturally associate to a p-summable regular spectral triple (A, H, D) a singularity structure, namely (A (A,H,D) , Ψ −∞ (A, H, D), H ∞ ). We shall denote this singularity structure by SS (A, H, D).

Abstract distributions.
We are now ready to formulate the notion of distributions for the more abstract setup. This generalization is motivated by the following observation.
Thus to complete the proof it suffices to show that the limit in the above assumption exists. Again since D ′ (M ) is a Montel space, it is enough to check the limit by evaluation on test-functions (smooth densities on M ). We fix a Riemannian density to trivialize the density bundle and observe that given any function f ∈ C ∞ (M ) there is a smoothing operator T such that T (1) = f 2 and therefore, This Proposition inspires the following definition.

Some functoriality considerations
We wish to develop a category with objects as singularity structures (A, X, Y ) as defined in 3.11 and suitable morphisms between them. To include the situation of the abstract distributions described in the previous section, we shall for the rest of the paper assume that X is a Fréchet bimodule over A. The first choice of such morphisms is rather straightforward and we describe them below.
Definition 4.1. Given two singularity structures (A j , X j , Y j ) j = 1, 2, a direct morphism α : (1) A morphism of filtered algebras α a : A 1 → A 2 . This provides X 2 and Y 2 with (pullback) A 1 module structures. (2) A tame linear map α r : X 1 → X 2 which is an A 1 bimodule map.
(3) A tame linear map α l : Y 2 → Y 1 which is also a left-A 1 module map. We call the direct morphism α monic if α a as well as the map induced by it on the associated graded algebras is injective,the map α r is injective and α l is surjective. In addition, for monic direct morphisms we require that the maps α l and α r have bounded order of tameness.
The above definition guarantees that given a direct morphism α : This has the further property that it preserves the Reg maps as well as distributions, α * : D A 2 (X 2 , Y 2 ) → D A 1 (X 1 , Y 1 ). Then the traditional functoriality of wavefront sets follows from the condition that α l and α r have bounded order of tameness and in this case where α 0 : Gr(A 1 ) 0 → Gr(A 2 ) 0 is the map on the 0-component of the associated graded algebras. Let us consider a rather simple example of functoriality properties of singularity structures relevant in classical analysis. Let (A, X, Y ) be a singularity structure and ρ : A ′ → A a morphism of filtered algebras. By pulling back modules under ρ we obtain a new singularity structure (A ′ , ρ * X, ρ * Y ) where we shall suppress ρ from the notation and write (A, X, Y ) instead for simplicity. Now ρ induces a mapρ : The inclusion C ∞ (M ) ֒→ Ψ ∞ (M ) as multiplication operators for instance provides the well known fact that the wavefront set of a distribution projects to its singular support.

Correspondence morphism.
Although the direct morphism defined in the previous section gives a nice functorial way of assigning wavefronts and regularity to abstract distributions, it turns out to be a rather restrictive concept as not all useful morphisms satisfy all the constraints of Definition 4.1. We shall here briefly describe a more general notion of correspondence and provide some simple examples. This should be thought of as a sort of "Morita equivalence" in the category Sing.

Definition 4.2.
A correspondence between two singularity structures (A j , X j , Y j ) j = 1, 2 is a span that is a third singularity structure (A, X, Y ) with two direct morphisms α j : We call a correspondence true correspondence provided the map α 1 is monic in the sense of definition 4.1 We shall abbreviate the data (A, X, Y ), α 1 , α 2 by A.
Correspondences can be composed since the category Sing d with direct morphisms has all pullbacks. In particular, composition of two monic/true correspondences is again a true correspondence. We shall refer to the category 3 of singularity structures with true spans as morphisms as Sing.

Definition 4.3. Let A be a correspondence between two singularity structures (
The pull-back of distributions provides an example of a correspondence as follows: The obvious morphism α 1 : Since f * is assumed to be injective, for an operator P ∈ Ψ ∞ f (M ) we can define α a 2 (P )(ψ) := f * −1 P (f * (ψ)). The following lemma implies that this correspondence morphism indeed represents the pull-back of a distribution in the usual sense. With the notations as above we have:

Pseudo-differential operators on groupoids
In this section we shall construct a morphism of singularity structures related to a Lie groupoid G over base M under the following two assumptions: (1) The space of arrows G (1) is a compact manifold. This condition can be weakened but we shall keep this assumption for simplicity of exposition.
(2) The anchor map on the Lie algebroid A(G) (described below) is surjective. This morphism β G presented below is used to connect the propagation of singularities on G to the propagation on the base manifold M . We begin by recalling a few facts about the groupoid calculus here.
Let G be a differential groupoid over the space of units M = G 0 . Let A(G) be its Lie-algebroid and we shall consider sections of A(G) as right-translation invariant vector-fields on G which are vertical with respect to the domain map d. Fix a section of the vertical density bundle D = |Λ|A(G) which provides a measure µ x on each fiber G x := d −1 x (x ∈ M ). As customary we denote by r the range map and by G x := r −1 x the fibers of r.
Given a vector bundle E over the space of units M a pseudo-differential operator on G of integer order m is a differentiable 4 family of classical pseudo-differential operators P := (P x ) x∈M where each P x is an element of Ψ m (G x : r * E) which is: (1) Uniformly supported: Let k x denote the distributional kernel of P x and define the support of P as Then P is said to be uniformly supported if the set {gh −1 |(g, h) ∈ supp P } is compact in G.
(2) Right invariant: For any element g ∈ G define a map U g : A family P is right-invariant iff U g P x = P y U g for all g where x = d(g) and y = r(g). The space of all pseudo-differential operators of order m is denoted by Ψ m (G : E). The regularizing elements are denoted by Ψ −∞ (G : E) := m Ψ m (G : E). The regularizing elements can be identified with the convolution algebra C ∞ c (G : End(E) ⊗ d * D). In case E is the trivial line bundle we shall write Ψ ∞ (G) for the corresponding algebra of pseudo-differential operators. The associated graded algebra can be identified with homogeneous sections of C ∞ (A * (G) : End(E)). An operator P is then called elliptic if σ(P ) is invertible away from the zero-section.
As mentioned before we will from now on assume that G is compact. To describe the singularity structure associated to G we shall recall a few preliminaries. Fix a choice of metric (nondegenerate, symmetric bilinear form) on A(G) to determine a trivialization of D and in addition fix a Hermitian inner product on E. This gives rise a pre-Hilbert-* module over C ∞ (M ) with field of Hilbert spaces given by L 2 (G x , r * E) for x ∈ M and sections given by restriction of continuous sections Γ(G : r * E) to each fiber G x . We shall denote the corresponding C(M ) C * -modules by E(M : E) and by E when the vector-bundle E is trivial.
The trivialization of D provides an involution on Ψ ∞ (G) . It follows that the elements of Ψ ∞ (G) are C ∞ (M )-linear (not necessarily bounded) regular operators on E. Further we fix a form positive self-adjoint operator D in Ψ ∞ (G). Then D is a regular operator on the C * -module E by the Woronowicz criterion (see, e.g., [22]) and hence defines a chain of Sobolev-modules E n . We set E ∞ := n E n and the norm of E n to provide the structure of a graded Fréchet space. It follows from the independence of the Sobolev chains E n from the operator D (see [22]) that E ∞ is a Fréchet-module over Ψ ∞ (G).
As G is assumed to be compact, the algebra Ψ −∞ (G) is a Fréchet module over Ψ ∞ (G) . The appropriate grading on Ψ −∞ (G) will be described below. In this case one obtain a Ψ(C * (G), C(M )) unbounded module (E, D). The extension to equivariant vector-bundles discussed below is analogous. We are now ready to define SS(G) as mentioned above.

5.1.
Vector representation on the units and the associated singularity structure. The algebra Ψ ∞ (G) consists of longitudinal operators on fibers G x of the domain map d. The vector representation assigns to each such operator a transversal operator, that is an operator on the base manifold M . This assignment gives a direct morphism from SS(G) to the singularity structure on M . First we recall some basics about the vector representation. (See [20] for more details.) Recall that the trivialization of the density bundle D chosen above also provides a norm on Ψ −∞ (G) given by Any representation of Ψ −∞ (G) that satisfies π(P ) ≤ P 1 extends uniquely to a bounded operator on Ψ ∞ (G), where by bounded one means that Ψ 0 (G) acts by bounded operators. We shall only consider representations π of Ψ −∞ (G) that are non-degenerate, that is π(Ψ −∞ (G))H = H ∞ is dense in H and the above condition on the norms hold. For such representations any self-adjoint elliptic operator Q of order 1 gives rise to an essentially self-adjoint operator π(Q). In the standard way one can define Sobolev space completions of H ∞ using powers of Q and such Sobolev spaces do not depend on the choice of the elliptic operator Q. Thus consider H ∞ with the graded Fréchet space structure provided by these Sobolev norms.
There is an obvious action of Ψ ∞ (G : E) on C ∞ (M : E) called the vector representation, obtained by setting P (f ) • r = P (f • r). We shall represent this action by π 0 . When the space of units is a compact manifold (possibly with corners) then (Ψ ∞ (G : E), Ψ −∞ (G : E), C ∞ (M )) provides an example of a singularity structure. In case E has the structure of an equivariant vector-bundle this gives rise to a representation of Ψ ∞ (G) as described below [20,14].
Let us consider a special case of vector bundle. An equivariant vector bundle (V, ρ) on a groupoid G is a vector-bundle V over the units M provided with a bundle isomorphism ρ : d * V → r * V such that ρ(gh) = ρ(g)ρ(h). Given an equivariant vector bundle (V, ρ) there is an action of Ψ ∞ (G) on C ∞ (M : V ) defined by identifying C ∞ c (G) ⊗ V x ≃ C ∞ (G : d * V ) and hence defining a map T ρ : Ψ ∞ (G) → Ψ ∞ (G, V ) given by T ρ (P ) := ρ(P ⊗ 1)ρ −1 and thus defining an action on C ∞ (M : V ) by π ρ := π 0 • T ρ . In fact analogous to Definition 5.1 we can define a singularity structure SS(G, V ) := (Ψ ∞ (G), Ψ −∞ (G), E ∞ (G, V )). All the results in this section generalize to the case of equivariant vector bundles, we shall however only formulate the case of a trivial vector bundle V and refer to the map π 0 as π.
As mentioned above we assume that the anchor map q : A(G) → T M is surjective. This provides a natural measure on M such that the vector representation becomes a bounded * representation. In addition it provides a direct morphism from SS(G) to the natural singularity structure on M .
Lemma 5.2. Let G be a compact Lie groupoid such that the anchor map q : A(G) → T M is surjective. Then the following hold: (1) There is a measure on M such that the vector representation gives rise to a bounded * representation of Ψ −∞ (G) on L 2 (M, V ). (2) There is a form-positive elliptic self-adjoint operator in Ψ 1 (G) whose image π ρ is an ordinary elliptic essentially self-adjoint operator of order 1.
Proof. Since we have fixed a metric on A(G) it identifies the complement ker(q) ⊥ with T M and hence provides a Riemannian metric on M . We shall denote the measure obtained from this metric by µ M . In addition the surjectivity of the anchor map implies that for any x ∈ M the range map r : G x → M is a submersion. Therefore the Riemannian metric on G x induced from A(G) splits into a tensor product of a density on fibers of r, that is along the submanifolds G y x := d −1 x ∩ r −1 y and a normal density µ M . For a smooth function φ ∈ C ∞ (M ) integration along the fiber gives Therefore (on the connected component of x ∈ M ) we choose the measure vol(G y x )dµ M and it is clear that on the corresponding L 2 (M, V ) the representation of Ψ −∞ (G) is a bounded * representation.
To show that there is an elliptic operator in the image of the vector representation of Ψ ∞ (G) one observes that a d-vertical right-invariant vector field X on G can be considered as an element in Ψ 1 (G). If s is the section of A(G) associated to the right-invariant vector field X then the vector representation π(X) equals the action of the vector field q(s). Since the anchor map q is assumed to be surjective, all vector fields and hence all differential operators on M lie in the range of the vector representation π. In fact these can be obtained from geometrical differential operators on G. (This assertion is equivalent to the fact that the geometrical differential operators on G, that is the space of family of right-invariant d-vertical differential operators on G x , can be identified with the symmetric algebra of Γ(M : A(G)).) In particular the bilinear form on A(G) gives a right invariant family of metrics on G x as x varies in M and the corresponding Laplace operators ∆ x is a family of elliptic self-adjoint operators on the module E and π(∆ x ) is an elliptic differential operator. We could consider the from-positive operator D = √ ∆ x + c. It follows that π(D) is an elliptic pseudo-differential operator on M as desired.
By applying results from [14] one can now construct the desired direct morphism SS(G) → SS(M ). Proof. Our first goal is to show that the vector-representation of an element P ∈ Ψ ∞ (G) is in fact a classical pseudo-differential operator on M of the same order, there by providing a map of filtered algebras π : Ψ ∞ (G) → Ψ ∞ (M ) It is clear from Lemma 5.2 that H = L 2 (M ) is a bounded * representation and by [14] the Sobolev spaces H s defined from a positive elliptic operator D ∈ Ψ 1 (G) over a bounded representation are independent of the choice of D. Furthermore it can be shown that any P ∈ Ψ m (G) provides a bounded operator π(P ) : H s → H s−m . From 5.2 it follows that π(D) being an elliptic pseudo-differential operator defines the usual Sobolev spaces, that is H s = H s (M ). By a classical theorem of Kohn-Nirenberg [12] any continuous operator L : C ∞ (M ) → C ∞ (M ) whose iterative commutators with any given finite set of vector fields Y j 0 ≤ j ≤ n has the property that However, verifying that π(P ) is a classical pseudo-differential operator requires the theory of Fourier integral operators. Briefly, on regularizing operators the vector representation π : Typically, given a self-adjoint elliptic operator Q ∈ Ψ 1 (M ) on a closed manifold an action of R on Ψ ∞ (M ) is given by t · P = P t = e itQ P e −itQ . The standard Egorov theorem says that P t is also a pseudo-differential operator whose principal symbols corresponds to the symplectic Hamiltonian flow of the symbol of P with Hamiltonian σ(Q). Such an action naturally extend to case of families of right invariant operators on a groupoid.
In general to obtain such an R action on a spectral-triple (A, H, D) by conjugation e it|D| certain conditions need to be imposed. Although the simple condition limsup n Log δ n (a) n < ∞, insures the convergence, it does not guarantee a priori that the resulting operator is a pseudo-differential operator.
In the following section we shall work only with commutative examples where actions are generated by an element of A.
6.1. The commutative case. The classical theorems about the propagation of singularities correspond to examples of singularity structures (A, X, Y ) such that the algebra B = A 0 /A −1 is a nice commutative algebra and the generator a of the action is a derivation on B through the commutator A 0 ∋ a 0 → [a 0 , a]. A prototypical example is provided by a Weyl algebra A in the sense of Guillemin [7].
Let C be a separable C * -algebra. In what follows, B(H) and K(H) denote the space of bounded and compact C-linear operators on a Hilbert C-module H, respectively. We shall suppress the mention of the algebra C in the notation. Definition 6.2. A singularity structure (A, X, Y ) is called commutative (over the C * -algebra C) if the algebra A satisfies the following conditions: (a) The algebra A = i∈Z A i is a unital *-algebra of (possibly unbounded) regular operators on a separable Hilbert C-module H. (b) A 0 ⊂ B(H) and A −1 ⊂ K(H). (c) The bracket reduces the filtration order by 1, that is The latter two conditions together imply that B = A 0 /A −1 is a unital commutative subalgebra of the Calkin algebra Q(H) := B(H)/K(H). The C * -completionB of B is a commutative C * algebra and henceB = C(N ) for a compact Hausdorff space N . We shall further assume that (d) The space N is a compact manifold, B = C ∞ (N ), and the commutator with an element a ∈ A 1 defines a smooth vector field on N . More precisely, given a ∈ A 1 there exists a unique vector field V a ∈ X(N ) such that: σ([a 0 , a]) = V a (σ(a 0 )).
Note that it follows from (c) above that the expression on the left hand side of the above equation is well defined.
Theorem 6.3. Let (A, X, Y ) be a commutative singularity structure and let a ∈ A 1 generate an action on A, X, Y . Let µ t be the flow generated by the vector field V a . Then for any φ ∈ P T (X, Y ), Proof. Set φ t = φ · t for φ ∈ PT(X, Y ). For any φ ∈ P T (X, Y ), abbreviate WF A (φ) by I φ . Let P ∈ A 0 , then it follows that: We shall now note that various propagation results can be explained in this framework. . This is of course well known when φ = Θ u by [11] as one clearly has Θ u · t = Θ e itD u which is the solution to the Cauchy problem above. If one considers maps in PT(M ) as generalized functions extending the distributions, then as a consequence of the singularity structure theorem a propagation of singularities of generalized functions also follows. 6.2. Given a groupoid G satisfying the conditions in Section 5 we observe that SS(G) is a commutative singularity structure over the algebra C(M ) where M is the space of units. Given an elliptic order one operator D ∈ Ψ 1 (G) the fiberwise Egorov theorem on G x gives an action on P ∈ Ψ ∞ (G) by P → e itD P e −itD := (e itDx P x e −itDx ) x . As in the previous section this defines an R action on SS(G). Proof. We need to show that π(e itD P e −itD ) = e itπ(D) π(P )e −itπ(D) .
For where g is also the unique solution to some Cauchy problem, d dt u(x, t) = iDu(x, t) u(−, 0) = f Pulling back with r these equations and the initial data one obtains the desired equality by appealing to the uniqueness of the solution to the mentioned Cauchy problem.
A consequence of 5.4 is that it relates the dynamics of singularities under longitudinal propagation of the distributions r * Θ u as seen in the symbol space C ∞ (A * (G) \ 0) with that of the distribution u under the transverse action in vector representation as claimed.