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Pseudodifferential Analysis on Continuous Family Groupoids
Abstract
We study properties and representations of the convo-lution algebra and the algebra of pseudodifferential operators asso-ciated to a continuous family groupoid. We show that the study of representations of the algebras of pseudodifferential operators of or-der zero completely reduces to the study of the representations of the ideal of regularizing operators. This recovers the usual boundedness theorems for pseudodifferential operators of order zero. We prove a structure theorem for the norm completions of these algebras asso-ciated to groupoids with invariant filtrations. As a consequence, we obtain criteria for an operator to be compact or Fredholm. We end with a discussion of the significance of these results to the index the-ory of operators on certain singular spaces. For example, we give a new approach to the question of the existence of spectral sections for operators on coverings of manifolds with boundary. We expect that our results will also play a role in the analysis on more general singular spaces.
... 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 . ...
... 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 . ...
... 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. ...
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.
... Actually, an alternative is to define it through the tangent groupoid of Connes, which was originally defined for the groupoid of a smooth manifold and later extended to the case of continuous family groupoids ( [20,13]). The tangent groupoid of a Lie groupoid G ⇒ G (0) is a Lie groupoid ...
... See [20,25,19,13,30] for a detailed presentation of pseudodifferential calculus on groupoids. ...
... extends to the C * -closures of the algebras and the assertion about the invertibility modulo compact operators amounts to the exactness of the sequence [13]: ...
For every connected manifold with corners we use a homology theory called conormal homology, defined in terms of faces and incidences and whose cycles correspond geometrically to corner's cycles. Its Euler characteristic (over the rationals, dimension of the total even space minus the dimension of the total odd space), , is given by the alternated sum of the number of (open) faces of a given codimension. The main result of the present paper is that for a compact connected manifold with corners X given as a finite product of manifolds with corners of codimension less or equal to three we have that 1) If X satisfies the Fredholm Perturbation property (every elliptic pseudodifferential b-operator on X can be perturbed by a b-regularizing operator so it becomes Fredholm) then the even Euler corner character of X vanishes, i.e. . 2) If the even Periodic conormal homology group vanishes, i.e. , then X satisfies the stably homotopic Fredholm Perturbation property (i.e. every elliptic pseudodifferential b-operator on X satisfies the same named property up to stable homotopy among elliptic operators). 3) If is torsion free and if the even Euler corner character of X vanishes, i.e. then X satisfies the stably homotopic Fredholm Perturbation property. For example for every finite product of manifolds with corners of codimension at most two the conormal homology groups are torsion free. The main theorem behind the above result is the explicit computation in terms of conormal homology of the theory groups of the algebra of b-compact operators for X as above. Our computation unifies the only general cases covered before, for codimension zero (smooth manifolds) and for codimension 1 (smooth manifolds with boundary).
... The point is that from the viewpoint of noncommutative geometry the algebra of ψDO on a stratified manifold is related to a certain groupoid (see [11,12]). Moreover, the group Ell(X) is related to the K-group of the groupoid C *algebra [13]. The Baum-Connes conjecture claims that the latter K-group is isomorphic to the topological K-group of the classifying space of the groupoid (see [14]). ...
... To this end, consider the map (see (13)) ...
... For small h, this family is invertible for all ξ. (This follows from the boundedness of the support of 1 − u and since (14) is uniform for const > λ ≥ 0 if in (13) we replace rξ by (r + λ)ξ). We obtain, by construction, ...
We find the stable homotopy classification of elliptic operators on stratified manifolds. Namely, we establish an isomorphism of the set of elliptic operators modulo stable homotopy and the K-homology group of the singular manifold. As a corollary, we obtain an explicit formula for the obstruction of Atiyah--Bott type to making interior elliptic operators Fredholm.
... Our main motivation to study gauge-invariant families comes from analysis and spectral theory on non-compact manifolds. For example, the Fredholmness of an elliptic differential operator on a manifold with a uniform structure at infinity is controlled by (possibly several) gauge-invariant families of elliptic differential operators, see [18,20,21]. This is a situation that arises when one studies the Dirac operator on an S 1 -manifold M , if we desingularize the action of S 1 by replacing the original metric g with φ −2 g, where φ is the length of the infinitesimal generator X of the S 1 -action. ...
... ⊗K follows, for example, from the results of [18]. In general, this local isomorphism gives ...
... obtained using the results of [18]. The operator A (or, rather, the family of operators A = (A b )) has an invertible principal symbol, and hence the family T = (T b ), G)) be the boundary map in the K-theory exact sequence ...
We define the gauge-equivariant index of a family of elliptic operators invariant with respect to the free action of a family \GR \to B of Lie groups (these families are called ``gauge-invariant families'' in what follows). If the fibers of \GR \to B are simply-connected and solvable, we compute the Chern character of the gauge-equivariant index, the result being given by an Atiyah-Singer type formula that incorporates also topological information about the bundle \GR \to B. The algebras of invariant pseudodifferential operators that we study, \Psm {\infty}Y and \PsS {\infty}Y, are generalizations of ``parameter dependent'' algebras of pseudodifferential operators (with parameter in ), so our results provide also an index theorem for elliptic, parameter dependent pseudodifferential operators. We apply these results to study Fredholm boundary conditions on a simplex.
... A number of papers are relevant to this construction in the context where X = G, e.g. [25,26,27,33,34,32,35,41,43]. A version of our construction under special assumptions -including the assumptions that G 0 is a proper, G-compact, G-space and that X is a fiber bundle over G 0 with compact smooth manifold as fiber -is proved in [46]. ...
... They also consider the adiabatic groupoid associated with a Lie groupoid, an example of which is the tangent groupoid as defined by Connes (in which case, the Lie groupoid is M × M ). This construction remains valid ( [25]) if the Lie groupoid is replaced by a continuous family groupoid. ...
... Pseudodifferential analysis on continuous family groupoids is studied by Lauter, Monthubert and Nisor in [25]. As described above, the Lie algebroid plays a fundamental role. ...
The paper constructs the analytic index for an elliptic pseudodifferential family of L^{m}_{\rho,\de}-operators invariant under the proper action of a continuous family groupoid on a G-compact, G-space.
... Let F ⊂ M be a closed subspace which is saturated, by which we mean that s −1 (F ) = r −1 (F ), and set O := M \ F . Then G F is a continuous family groupoid [44,27]. By restricting over F , we get the F -indicial symbol map: ...
... We write Σ F for Σ 0 F . This gives the short exact sequence of C * -algebras [27]: ...
... The analogy to the commutative case is that Σ F is the noncommutative cosphere bundle, relative to F , and T F M is the noncommutative tangent bundle, relative to F , associated with G ⇒ M , see also [27,Section 5]. Denote by ∂ : K 1 (Σ F ) → K 0 (C * (G O )) the connecting "index" map in the K-theory sixterm exact sequence associated to the short exact sequence of the full symbol map (4). ...
We review recent progress regarding the index theory of operators defined on non-compact manifolds that can be modeled by Lie groupoids. The structure of a particular type of almost regular foliation is recalled and the construction of the corresponding accompanying holonomy Lie groupoid. Using deformation groupoids, K-theoretical invariants can be defined and compared. We summarize how questions in index theory are addressed via the geometrization made possible by the use of deformation groupoids. The discussion is motivated by examples and applications to degenerate PDE’s, diffusion processes, evolution equations and geometry.
... To capture the defect of Fredholmness of elliptic b-operators on X, we may introduce the algebra of full, or joint, symbols A F [12]. If F 1 denotes the set of closed boundary hypersurfaces of X, then the full symbol map is the * -homomorphism given by: ...
Given a connected manifold with corners X of any codimension there is a very basic and computable homology theory called conormal homology defined in terms of faces and orientations of their conormal bundles, and whose cycles correspond geometrically to corner's cycles, these conormal homology groups are denoted by . Using our previous works we define an index morphism for X a manifold with corners of codimension less or equal to three and called here the even conormal index morphism. In the case that X is compact and connected and D is an elliptic pseudodifferential operator in the associated calculus of X we know, by our previous works and other authors works, that, up to adding an identity operator, D can be perturbed (with a regularizing operator in the calculus) to a Fredholm operator iff (where is the principal symbol class) vanishes in the even conormal homology group . The main result of this paper is the explicit computation of the even and odd conormal index morphisms for X a manifold with corners of codimension less or equal to three. The coefficients of the conormal corner cycles are given in terms of some suspended Atiyah-Singer indices of the maximal codimension faces of X and in terms of some suspended Atiyah-Patodi-Singer indices of the non-maximal codimension faces of X. As a corollary we give a complete caracterization to the obstruction of the Fredholm perturbation property for closed manifolds with corners of codimension less or equal to three in terms of the above mentioned indices of the faces, this allows us as well to give such a characterization in terms of the respective topological indices.
... Much like we can associate a Lie algebra with any Lie group, with any differentiable groupoid G is associated a Lie algebroid (see, e.g., [46]). In particular, this is important in the context of the pseudodifferential calculus on groupoids (see, e.g., [5,25,44,57,59,73]). Recall that a Lie algebroid structure on a vector bundle A over a manifold M is given by (i) A Lie bracket on the space of sections C 8 pM, Aq. ...
This paper studies the infinitesimal structure of Carnot manifolds. By a Carnot manifold we mean a manifold together with a subbundle filtration of its tangent bundle which is compatible with the Lie bracket of vector fields. We introduce a notion of differential, called Carnot differential, for Carnot manifolds maps (i.e., maps that are compatible with the Carnot manifold structure). This differential is obtained as a group map between the corresponding tangent groups. We prove that, at every point, a Carnot manifold map is osculated in a very precise way by its Carnot differential at the point. We also show that, in the case of maps between nilpotent graded groups, the Carnot differential is given by the Pansu derivative. Therefore, the Carnot differential is the natural generalization of the Pansu derivative to maps between general Carnot manifolds. Another main result is a construction of an analogue for Carnot manifolds of Connes' tangent groupoid. Given any Carnot manifold (M,H) we get a smooth groupoid that encodes the smooth deformation of the pair to the tangent group bundle GM. This shows that, at every point, the tangent group is the tangent space in a true differential-geometric fashion. Moreover, the very fact that we have a groupoid accounts for the group structure of the tangent group. Incidentally, this answers a well-known question of Bella\"iche.
... In this appendix, we recall some classical results on families of pseudodifferential operators. We refer the reader to [26,27,28,29,30,31,6]. ...
We define and study the index map for families of G-transversally elliptic operators and introduce the multiplicity for a given irreducible representation as a virtual bundle over the base of the fibration. We then prove the usual axiomatic properties for the index map extending the Atiyah-Singer results [1]. Finally, we compute the Kasparov intersection product of our index class against the K-homology class of an elliptic operator on the base. Our approach is based on the functorial properties of the intersection product, and relies on some constructions due to Connes-Skandalis and to Hilsum-Skandalis.
We construct an algebra of pseudodifferential operators on each groupoid in a class that generalizes differentiable groupoids to allow manifolds with corners. We show that this construction encompasses many examples. The subalgebra of regularizing operators is identified with the smooth algebra of the groupoid, in the sense of non-commutative geometry. Symbol calculus for our algebra lies in the Poisson algebra of functions on the dual of the Lie algebroid of the groupoid. As applications, we give a new proof of the Poincare-Birkhoff-Witt theorem for Lie algebroids and a concrete quantization of the Lie-Poisson structure on the dual A* of a Lie algebroid.
Introduction. Let K be a commutative ring with ulnit, and let R be a commutative unitary K-algebra. We shall be concerned with variously defined cohomology theories based on algebras of differential forms, where R plays the role of a ring of functions. Let TR be the Lie algebra of the K-derivations of R, and let E(TR) be the exterior algebra over R of TR. We can form HoomR(E(TR), R) and define on it the usual formal differentiation. If R is the ring of functions on a C?-rmanifold then the elements of TR are the differentiable tangent vector fields, and the complex HomR(E(TR),R) is naturally isomorphic to the usual de Rham complex of differential forms. In [5, ?? 6-9] the complex HoMR(E(TR), R) is studied. It is shown that if K is a field contained in R, and if either R is an integral domain finitely ring-generated over K and TR is R-projective, or R is a field, then the homology of this complex may be identified with Extv(R, R), for a suitably defined ring V. ??1-6 of the present work are primarily a straightforward generalization of the results of this portion of [5] to the case in which K and R are arbitrary (commutative) rings. In making this generalization we are led naturally to replace TR by an arbitrary Lie algebra with an R-module structure which is represented as derivations of R and which satisfies certain additional properties satisfied by TR. We give these properties in ?2. Lis essentially a quasi-Lie algebra as defined in [3]. The precise definition given corresponds to that of a d-Lie ring given in [8], where also the cohomology based on HomR(E(L), A) is defined. In ?2 we define an associative algebra V of universal differential operators generated by R and L. In case L operates trivially on R, V is the usual universal enveloping algebra of the R-Lie algebra L. In ?3 we prove a Poincare-BirkhoffWitt theorem for V. In ?4 we show that if Lis R-projective then for any V-module A we may identify the cohomology based on HomR(E(L), A) with Extv(R, A), which we denote by HR(L, A). In particular, the de Rham cohomology of a C manifold is thus identified with an Extv(R, R).
We compute the length of the C*-algebra generated by the algebra of b-pseudodifferential operators of order 0 on compact manifolds with corners.