Elliptic Theory on Manifolds with Corners: I. Dual Manifolds and Pseudodifferential Operators
V. E. Nazaikinskii, A. Yu. Savin, B. Yu. Sternin
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21 pages, 2 figures. Submitted to the proceedings of the Conference "C*-algebras and elliptic theory. II" (Banach Center, Bedlewo, Poland, January 23--28, 2006)
In this first part of the paper, we define a natural dual object for manifolds with corners and show how pseudodifferential calculus on such manifolds can be constructed in terms of the localization principle in C*-algebras. In the second part, these results will be applied to the solution of Gelfand's problem on the homotopy classification of elliptic operators for the case of manifolds with corners.
We build a longitudinally smooth differentiable groupoid associated to any manifold with corners. The pseudodifferential calculus on this groupoid coincides with the pseudodifferential calculus of Melrose (also called b-calculus). We also define an algebra of rapidly decreasing functions on this groupoid; it contains the kernels of the smoothing operators of the (small) b-calculus.
This paper is concerned with pseudodifferential calculus on manifolds with fibred corners. Following work of Connes, Monthubert, Skandalis and Androulidakis, we associate to every manifold with fibred corners a longitudinally smooth groupoid which algebraic and differential structure is explicitely described. This groupoid has a natural geometric meaning as a holonomy groupoid of singular foliation, it is a singular leaf space in the sense of Androulidakis and Skandalis. We then show that the associated compactly supported pseudodifferential calculus coincides with Mazzeo and Melrose's phi-cal…
Let L^\star be a filtered algebra of abstract pseudodifferential operators equipped with a notion of ellipticity, and T^\star be a subalgebra of operators of the form P_1AP_0, where P_0 and P_1 are two projections. The elements of L^\star act as linear continuous operators in certain scales of abstract Sobolev spaces, the elements of the subalgebra in the corresponding subspaces determined by the projections. We study how the ellipticity in L^\star descends to T^\star, focusing on parametrix construction, Fredholm property, and homogeneous principal symbols. Applications concern SG-pseudodiffe…
We compute the Hochschild, cyclic, and periodic cyclic homology groups of algebras of families of Laurent complete symbols on manifolds with corners. We show in particular that the spectral sequence associated with Hochschild homology degenerates at $E^2$ and converges to Hochschild homology. As a byproduct, we deduce an identification of the space of residue traces on fibrations by manifolds with corners. In the process, we prove several general results about algebras of complete symbols on manifolds with corners.
In this paper, we introduce a parametric pseudodifferential calculus on noncommutative $n$-tori which is a natural nest for resolvents of elliptic pseudodifferential operators. Unlike in some previous approaches to parametric pseudodifferential calculi, our parametric pseudodifferential calculus contains resolvents of elliptic pseudodifferential operators that need not be differential operators. As an application we show that complex powers of positive elliptic pseudodifferential operators on noncommutative $n$-tori are pseudodifferential operators. This confirms a claim of Fathi-Ghorbanpour-K…
In previous papers (arxiv:math/0612370 and arxiv:0909.1342) we defined the C*-algebra and the longitudinal pseudodifferential calculus of any singular foliation (M,F). Here we construct the analytic index of an elliptic operator as a KK-theory element, and prove that the same element can be obtained from an "adiabatic foliation" TF on $M \times \R$, which we introduce here.
We prove the spectral invariance of the algebra of classical pseudodifferential boundary value problems on manifolds with conical singularities in the Lp-setting. As a consequence we also obtain the spectral invariance of the classical Boutet de Monvel algebra of zero order operators with parameters. In order to establish these results, we show the equivalence of Fredholm property and ellipticity for both cases.
We present a calculus of pseudodifferential operators that contains both usual parameter-dependent operators -- where a real parameter τ enters as an additional covariable -- as well as operators not depending on τ. Parameter-ellipticity is characterized by the invertibility of three associated principal symbols. The homogeneous principal symbol is not smooth on the whole co-sphere bundle but only admits directional limits at the north-poles, encoded by a principal angular symbol. Furthermore there is a limit-family for τ\to+\infty. Ellipticity permits to construct parametrices that are invers…
Starting out from a new description of a class of parameter-dependent pseudodifferential operators with finite regularity number due to G. Grubb, we introduce a calculus of parameter-dependent, poly-homogeneous symbols whose homogeneous components have a particular type of point-singularity in the covariable-parameter space. Such symbols admit intrinsically a second kind of expansion which is closely related to the expansion in the Grubb-Seeley calculus and permits to recover the resolvent-trace expansion for elliptic pseudodifferential oerators originally proved by Grubb-Seeley. Another appli…
We compute $K$-theory invariants of algebras of pseudodifferential operators on manifolds with corners and prove an equivariant index theorem for operators invariant with respect to an action of $\R^k.$ We discuss the relation between our results and the $η$-invariant.
We introduce a notion of ellipticity of complexes of linear pseudodifferential operators acting on sections of $A$-Hilbert bundles over smooth manifolds, $A$ being a $C^*$-algebra. We prove that the cohomology groups of an $A$-elliptic pseudodifferential complex in finitely generated projective $A$-Hilbert bundles over a compact manifold are norm complete finitely generated $A$-modules if the images of the associated Laplacians are closed. This establishes a Hodge theory for these structures.
Let $X$ be a smooth compact manifold with corners which has two embedded boundary hypersurfaces $\partial_0 X , \partial_1 X$, and a fiber bundle $ϕ:\partial_0 X \to Y$ is given. By using the method of blowing up, we define a pseudodifferential culculus $Ψ^* _{Φ,b} (X)$ generalizing the $Φ$-calculus of Mazzeo and Melrose and the (small) $b$-calculus of Melrose. We discuss the Fredholm condition of such operators and prove the relative index theorem. And as its application, the index theorem of "non-closed" $\mathbb{Z}/k$ - manifolds is proved.
We establish the stable homotopy classification of elliptic pseudodifferential operators on manifolds with corners and show that the set of elliptic operators modulo stable homotopy is isomorphic to the K-homology group of some stratified manifold. By way of application, generalizations of some recent results due to Monthubert and Nistor are given.
We study the complex powers $A^{z}$ of an elliptic, strictly positive pseudodifferential operator $A$ using an axiomatic method that combines the approaches of Guillemin and Seeley. In particular, we introduce a class of algebras, ``extended Weyl algebras,'' whose definition was inspired by Guillemin's paper on the subject. An extended Weyl algebra can be thought of as an algebra of ``abstract pseudodifferential operators.'' Many algebras of pseudodifferential operators are extended Weyl algebras. Several results typical for algebras of pseudodifferential operators (asymptotic completeness, co…
This paper is the 2nd part of a two-paper series whose aim is to give a detailed description of Connes' pseudodifferential calculus on noncommutative $n$-tori, $n\geq 2$. We make use of the tools introduced in the 1st part to deal with the main properties of pseudodifferential operators on noncommutative tori of any dimension $n\geq 2$. This includes the main results mentioned in the original notes of Connes and Baaj. We also obtain further results regarding action on Sobolev spaces, spectral theory of elliptic operators, and Schatten-class properties of pseudodifferential operators of negativ…
We construct algebras of pseudodifferential operators on a continuous family groupoid G that are closed under holomorphic functional calculus, contain the algebra of all pseudodifferential operators of order 0 on G as a dense subalgebra, and reflect the smooth structure of the groupoid G, when G is smooth. As an application, we get a better understanding on the structure of inverses of elliptic pseudodifferential operators on classes of non-compact manifolds. For the construction of these algebras closed under holomorphic functional calculus, we develop three methods: one using two-sided semi-…
Given a Lie group $G$ of quantized canonical transformations acting on the space $L^2(M)$ over a closed manifold $M$, we define an algebra of so-called $G$-operators on $L^2(M)$. We show that to $G$-operators we can associate symbols in appropriate crossed products with $G$, introduce a notion of ellipticity and prove the Fredholm property for elliptic elements. This framework encompasses many known elliptic theories, for instance, shift operators associated with group actions on $M$, transversal elliptic theory, transversally elliptic pseudodifferential operators on foliations, and Fourier in…
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 g…
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 Poincaré-Birkhoff-Witt theorem for Lie algebroids and a concrete quantization …
We consider the operator algebra generated by pseudodifferential operators on a closed smooth surface and shift operator induced by a Morse--Smale diffeomorphism of this surface. Elements in this algebra are considered as operators in the scale of Sobolev spaces and the aim of this paper is to describe how Fredholm property of a given operator depends on the Sobolev smoothness exponent $s$.
This memoir is a summary of recent work, including collaborations with Erik van Erp, Christian Voigt and Marco Matassa, compiled for the "Habilitation à diriger des recherches". We present various different approaches to constructing algebras of pseudodifferential operators adapted to filtered and multifiltered manifolds and some quantum analogues. A general goal is the study of index problems in situations where standard elliptic theory is insufficient. We also present some applications of these constructions. We begin by presenting a characterization of pseudodifferential operators on filter…
A C*algebra A generated by a class of zero-order classical pseudodifferential operator on a cylinder RxB, where B is a compact riemannian manifold, containing operators with periodic symbols, is considered. A description of the K-theory index map associated to the continuous extension to A of the principal-symbol map is given. That index map takes values in K_0 of the commutator ideal E of the algebra, which is isomorphic to Z^2. It maps the K_1-class of an operator invertible modulo E to the Fredholm indices of a pair of elliptic pseudodifferentail operators on SxB, where S denotes the circle…
We identify Melrose's suspended algebra of pseudodifferential operators with a subalgebra of the algebra of parametric pseudodifferential operators with parameter space $\R$. For a general algebra of parametric pseudodifferential operators, where the parameter space may now be a cone $Γ\subset\R^p$, we construct a unique ``symbol valued trace'', which extends the $L^2$-trace on operators of small order. This allows to construct various trace functionals in a systematic way. Furthermore we study the higher-dimensional eta-invariants on algebras with parameter space $\R^{2k-1}$. Using Clifford r…
We extend the classical regularity theorem of elliptic operators to maximally hypoelliptic differential operators. More precisely, given vector fields $X_1,\ldots,X_m$ on a smooth manifold which satisfy Hörmander's bracket generating condition, we define a principal symbol for \textit{any} linear differential operator. Our symbol takes into account the vector fields $X_i$ and their commutators. We show that for an arbitrary differential operator, its principal symbol is invertible if and only if the operator is maximally hypoelliptic. This answers affirmatively a conjecture due to Helffer and …
We define a class of boundary value problems on manifolds with fibered boundary. This class is in a certain sense a deformation between the classical boundary value problems and the Atiyah-Patodi-Singer problems in subspaces. The boundary conditions in this theory are taken as elements of the C^*-algebra generated by pseudodifferential operators and families of pseudodifferential operators in the fibers. We prove the Fredholm property for elliptic boundary value problems. This class also contains nonlocal boundary value problems of math.KT/0108107. We compute a topological obstruction (similar…
Given a manifold with boundary endowed with an action of a discrete group on it, we consider the algebra of operators generated by elements in the Boutet de Monvel algebra of pseudodifferential boundary value problems and shift operators acting on functions on the manifold and its boundary. Under certain conditions on the group action, we construct a Chern character for symbols of elliptic elements in this algebra with values in a de Rham type cohomology of the fixed point manifolds and obtain an index formula in terms of this Chern character.
We answer the question of when an invariant pseudodifferential operator is Fredholm on a fixed, given isotypical component. More precisely, let $Γ$ be a compact group acting on a smooth, compact, manifold $M$ without boundary and let $P \in ψ^m(M; E_0, E_1)$ be a $Γ$-invariant, classical, pseudodifferential operator acting between sections of two $Γ$-equivariant vector bundles $E_0$ and $E_1$. Let $α$ be an irreducible representation of the group $Γ$. Then $P$ induces by restriction a map $π_α(P) : H^s(M; E_0)_α\to H^{s-m}(M; E_1)_α$ between the $α$-isotypical components of the corresponding S…
We consider the index problem for a wide class of nonlocal elliptic operators on a smooth closed manifold, namely differential operators with shifts induced by the action of an isometric diffeomorphism. The key to the solution is the method of uniformization: We assign to the nonlocal problem a pseudodifferential operator with the same index, acting in sections of an infinite-dimensional vector bundle on a compact manifold. We then determine the index in terms of topological invariants of the symbol, using the Atiyah-Singer index theorem.
It is well known that elliptic operators on a smooth compact manifold are classified by K-homology. We prove that a similar classification is also valid for manifolds with simplest singularities: isolated conical points and fibered boundary. The main ingredients of the proof of these results are: an analog of the Atiyah-Singer difference construction in the noncommutative case and an analog of Poincare isomorphism in K-theory for our singular manifolds. As applications we give a formula in topological terms for the obstruction to Fredholm problems on manifolds with singularities and a formula …
This paper provides an E-theoretic proof of an exact form, due to E. Troitsky, of the Mischenko-Fomenko Index Theorem for elliptic pseudodifferential operators over a unital C*-algebra. The main ingredients in the proof are the use of asymptotic morphisms of Connes and Higson, vector bundle modification, a Baum-Douglas-type group, and a KK-argument of Kasparov.
An elliptic theory is constructed for operators acting in subspaces defined via odd pseudodifferential projections. Subspaces of this type arise as Calderon subspaces for first order elliptic differential operators on manifolds with boundary, or as spectral subspaces for self-adjoint elliptic differential operators of odd order. Index formulas are obtained for operators in odd subspaces on closed manifolds and for general boundary value problems. We prove that the eta-invariant of operators of odd order on even-dimesional manifolds is a dyadic rational number.
We show that the computation of the Fredholm index of a fully elliptic pseudodifferential operator on an integrated Lie manifold can be reduced to the computation of the index of a Dirac operator, perturbed by a smoothing operator, canonically associated, via the so-called clutching map. To this end we adapt to our framework ideas coming from Baum-Douglas geometric $K$-homology and in particular we introduce a notion of geometric cycles, that can be categorized as a variant of the famous geometric $K$-homology groups, for the specific situation here. We also define a comparison map between thi…
The goal of this paper is to construct a calculus whose higher indices are naturally elements in the twisted K-theory groups for Lie groupoids. Given a Lie groupoid $G$ and a $PU(H)$-valued groupoid cocycle, we construct an algebra of projective pseudodifferential operators. The subalgebra of regularizing operators identifies with the naturally associated smooth convolution algebra of the associated twisted gerbe. We develop the associated symbolic calculus, symbol short exact sequences and existence of parametrices. In particular the algebra of projective operators appears as a quantization o…
Boutet de Monvel's calculus provides a pseudodifferential framework which encompasses the classical differential boundary value problems. In an extension of the concept of Lopatinski and Shapiro, it associates to each operator two symbols: a pseudodifferential principal symbol, which is a bundle homomorphism, and an operator-valued boundary symbol. Ellipticity requires the invertibility of both. If the underlying manifold is compact, elliptic elements define Fredholm operators. Boutet de Monvel showed how then the index can be computed in topological terms. The crucial observation is that elli…
Cobordism invariance shows that the index, in K-theory, of a family of pseudodifferential operators on the boundary of a fibration vanishes if the symbol family extends to be elliptic across the whole fibration. For Dirac operators with spectral boundary condition, Dai and Freed \cite{dai-freed1} gave an explicit version of this at the level of the determinant bundle. Their result, that the eta invariant of the interior family trivializes the determinant bundle of the boundary family, is extended here to the wider context of pseudodifferential families of cusp type.
We prove index formulas for elliptic operators acting between sections of C*-vector bundles on a closed manifold. The formulas involve Karoubi's Chern character from K-theory of a C*-algebra to de Rham homology of smooth subalgebras. We show how they apply to the higher index theorem for coverings and to flat foliated bundles, and prove an index theorem for C*-dynamical systems associated to actions of compact Lie groups. In an Appendix we relate the pairing of odd K-theory and KK-theory to the noncommutative spectral flow and prove the regularity of elliptic pseudodifferential operators over …
For a $C^*$-algebra $A$ of compact operators and a compact manifold $M,$ we prove that the Hodge theory holds for $A$-elliptic complexes of pseudodifferential operators acting on smooth sections of finitely generated projective $A$-Hilbert bundles over $M.$ For these $C^*$-algebras, we get also a topological isomorphism between the cohomology groups of an $A$-elliptic complex and the space of harmonic elements. Consequently, the cohomology groups appear to be Banach spaces. We prove as well, that if the Hodge theory holds for a complex in the category of Hilbert $A$-modules and continuous adjo…
The paper constructs the analytic index for an elliptic pseudodifferential family of $L^{m}_{ρ,\de}$-operators invariant under the proper action of a continuous family groupoid on a $G$-compact, $C^{\infty,0}$ $G$-space.
We extend formulae which measure discrepancies for regularized traces on classical pseudodifferential operators to regularized trace cochains, regularized traces corresponding to 0-regularized trace cochains. This extension from 0-cochains to $n$-cochains is appropriate to handle simultaneously algebraic and geometric discrepancies/anomalies. Algebraic anomalies are Hochschild coboundaries of regularized trace cochains on a fixed algebra of pseudodifferential operators weighted by a fixed classical pseudodifferential operator with positive order and positive scalar leading symbol. In contrast,…
In the paper we consider the theory of elliptic operators acting in subspaces defined by pseudodifferential projections. This theory on closed manifolds is connected with the theory of boundary value problems for operators violating Atiyah-Bott condition. We prove an index formula for elliptic operators in subspaces defined by even projections on odd-dimensional manifolds and for boundary value problems, generalizing the classical result of Atiyah-Bott. Besides a topological contribution of Atiyah-Singer type, the index formulas contain an invariant of subspaces defined by even projections. Th…
We compute the $K$-theory of comparison $C^*$-algebra associated to a manifold with corners. These comparison algebras are an example of the abstract pseudodifferential algebras introduced by Connes and Moscovici \cite{M3}. Our calculation is obtained by showing that the comparison algebras are a homomorphic image of a groupoid $C^*$-algebra. We then prove an index theorem with values in the $K$-theory groups of the comparison algebra.
Families of representations of suitable Banach algebras provide a powerful tool in the study of the spectral theory of (pseudo)differential operators and of their Fredholmness. We introduce the new concept of an exhausting family of representations of a C*-algebra A. An {\em exhausting family} of representations of a C*-algebra A is a set F of representations of A with the property that every irreducible representation of A is weakly contained in some ϕ\in F. An exhausting family F of representations of A has the property that `"a \in A is invertible if, and if, ϕ(a) is invertible for any ϕ\in…
We prove the regularity of the $η$ function for classical pseudodifferential operators with Shubin symbols. We recall the construction of complex powers and of the Wodzicki and Kontsevich-Vishik functionals for classical symbols on $\mathbb{R}^{n}$ with these symbols. We then define the $ζ$ and $η$ functions associated to suitable elliptic operators. We compute the $K_{0}$ group of the algebra of zero-order operators and use this knowledge to show that the Wodzicki trace of the idempotents in the algebra vanishes. From this, it follows that the $η$ function is regular at 0 for any self-adjoint…
The aim of this paper is to show how zeta functions and excision in cyclic cohomology may be combined to obtain index theorems. In the first part, we obtain a local index formula for "abstract elliptic pseudodifferential operators" associated to spectral triples. This formula is notably well adapted when the zeta function has multiple poles. The second part is devoted to give a concrete realization of this formula by deriving an index theorem on the simple, but significant example of Heisenberg elliptic operators on a trivial foliation, which are in general non-elliptic but hypoelliptic. The l…
We define an analytic index and prove a topological index theorem for a non-compact manifold $M\_0$ with poly-cylindrical ends. We prove that an elliptic operator $P$ on $M\_0$ has an invertible perturbation $P+R$ by a lower order operator if an only if its analytic index vanishes. As an application, we determine the $K$-theory groups of groupoid $C^*$--algebras of manifolds with corners.
We investigate a quantization problem which asks for the construction of an algebra for relative elliptic problems of pseudodifferential type associated to smooth embeddings. Specifically, we study the problem for embeddings in the category of compact manifolds with corners. The construction of a calculus for elliptic problems is achieved using the theory of Fourier integral operators on Lie groupoids. We show that our calculus is closed under composition and furnishes a so-called noncommutative completion of the given embedding. A representation of the algebra is defined and the continuity of…
