C*-algebras and Elliptic Theory (original) (raw)
2006, Trends in Mathematics
The paper "Index Theory for Generalized Dirac Operators on Open Manifolds" by J. Eichhorn is devoted to the index theory on open manifolds. In the first part of the paper, a short review of index theory on open manifolds is given. In the second part, a general relative index theorem admitting compact topological perturbations and Sobolev perturbations of all other ingredients is established. V. Nazaikinskii and B. Sternin in the paper "Lefschetz Theory on Manifolds with Singularities" extend the Lefschetz formula to the case of elliptic operators on the manifolds with singularities using the semiclassical asymptotic method. In the paper "Pseudodifferential Subspaces and Their Applications in Elliptic Theory" by A. Savin and B. Sternin the method of so called pseudodifferential projectors in the theory of elliptic operators is studied. It is very useful for the study of boundary value problems, computation of the fractional part of the spectral AtiyahPatodiSinger eta invariant and analytic realization of topological K-groups with finite coefficients in terms of elliptic operators. In the paper "Residues and Index for Bisingular Operators" F. Nicola and L. Rodino consider an algebra of pseudo-differential operators on the product of two manifolds, which contains, in particular, tensor products of usual pseudo-differential operators. For this algebra the existence of trace functionals like Wodzickis residue is discussed and a homological index formula for the elliptic elements is proved. B. Bojarski and A. Weber in their paper "Correspondences and Index" define a certain class of correspondences of polarized representations of C *-algebras. These correspondences are modeled on the spaces of boundary values of elliptic operators on bordisms between two manifolds. In this situation an index is defined. The additivity of this index is studied in the paper. Noncommutative aspects of Morse theory: In the paper "New L2-invariants of Chain Complexes and Applications" by V.V. Sharko homotopy invariants of free cochain complexes and Hilbert complex are studied. These invariants are applied to calculation of exact values of Morse numbers of smooth manifolds. A. Connes and T. Fack in their paper "Morse Inequalities for Foliations" outline an analytical proof of Morse inequalities for measured foliations obtained by them previously and give some applications. The proof is based on the use of a twisted Laplacian. Riemannian aspects: The paper "A Riemannian Invariant, Euler Structures and Some Topological Applications" by D. Burghelea and S. Haller discusses a numerical invariant associated with a Riemannian metric, a vector field with isolated zeros, and a closed one form which is defined by a geometrically regularized integral. This invariant extends the ChernSimons class from a pair of two Riemannian metrics to a pair of a Riemannian metric and a smooth triangulation. They discuss a generalization of Turaevs Euler structures to manifolds with non-vanishing Euler characteristics and introduce the Poincare dual concept of co-Euler structures. The duality is provided by a geometrically regularized integral and involves the invariant mentioned above. Euler structures have been introduced because they permit to remove the ambiguities in the definition of the Reidemeister torsion. Similarly, co-Euler structures can be used to eliminate the metric dependence of
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A Riemannian Invariant, Euler Structures and Some Topological Applications
Trends in Mathematics
In this paper: (i) We define and study a new numerical invariant R(X, g, ω) associated with a closed Riemannian manifold (M, g), a closed one form ω and a vector field X with isolated zeros. When X = − grad g f with f : M → R a Morse function this invariant is implicit in the work of Bismut-Zhang. The definition of this invariant requires "geometric regularization". (ii) We define and study the sets of Euler structures and co-Euler structures of a based pointed manifold (M, x 0). When χ(M) = 0 the concept of Euler structure was introduced by V. Turaev. The Euler resp. co-Euler structures permit to remove the geometric anomalies from Reidemeister torsion resp. Ray-Singer torsion. (iii) We apply these concepts to torsion related issues, cf. Theorems 3 and 4. In particular we show the existence of a meromorphic function associated to a pair (M, e *), consisting of a smooth closed manifold and a co-Euler structure, defined on the variety of complex representations of the fundamental group of M whose real part is the Ray-Singer torsion (corrected). This function generalizes the Alexander polynomial for the complement of a knot.
Index theory of hypoelliptic operators on Carnot manifolds
2022
We study the index theory of hypoelliptic operators on Carnot manifolds-manifolds whose Lie algebra of vector fields is equipped with a filtration induced from sub-bundles of the tangent bundle. A Heisenberg pseudodifferential operator, elliptic in the calculus of van Erp-Yuncken, is hypoelliptic and Fredholm. Under some geometric conditions, we compute its Fredholm index by means of operator K-theory. These results extend the work of Baum-van Erp (Acta Mathematica '2014) for co-oriented contact manifolds to a methodology for solving this index problem geometrically on Carnot manifolds. Under the assumption that the Carnot manifold is regular, i.e. has isomorphic osculating Lie algebras in all fibres, and admits a flat coadjoint orbit, the methodology derived from Baum-van Erp's work is developed in full detail. In this case, we develope K-theoretical dualities computing the Fredholm index by means of geometric K-homology a la Baum-Douglas. The duality involves a Hilbert space bundle of flat orbit representations. Explicit solutions to the index problem for Toeplitz operators and operators of the form "∆ H`γ T " are computed in geometric K-homology, extending results of Boutet de Monvel and Baum-van Erp, respectively, from co-oriented contact manifolds to regular polycontact manifolds. The existence and the precise form of the geometric duality constructed for the Heisenberg calculus relies on the representation theory in the flat coadjoint orbits of the osculating Lie groupoid. We address the technical issue of constructing a Hilbert space bundle of representations associated to the flat coadjoint orbits via Kirillov's orbit method. The construction intertwines the index theory of Heisenberg operators to characteristic classes constructed from the Carnot structure further clarifying the two opposite spin c-structures appearing in Baum-van Erp's solution to the index problem on contact manifolds. Contents 1. Introduction 2. Summary of contents 3. Acknowledgements Part 1. Simply connected nilpotent Lie groups 4. Representation theory 5. The fine stratification of the spectrum 6. Flat orbits 7. Automorphisms and their action onĜ 8. The continuous trace algebra structure of the ideal of flat orbits 9. Examples of graded nilpotent Lie groups Part 2. Groupoids and twists 10. Lie groupoids and their C˚-algebras 11. A groupoid description of the flat orbits 12. Locally trivial bundles of nilpotent Lie groups 13. Nistor's Connes-Thom isomorphism and the ideal of flat orbits 14. The Thom-Connes isomorphism and the bundle of flat representations Part 3. Carnot manifolds and associated groupoids 15. Carnot manifolds 16. Examples of Carnot manifolds 17. The parabolic tangent groupoid Part 4. H-elliptic operators on Carnot manifolds 18. The pseudodifferential calculus of van Erp-Yuncken 19. H-ellipticity and the Rockland condition 20. The action of H-elliptic operators on certain Hilbert C˚-modules 21. K-theoretical invariants of H-elliptic operators Part 5. K-homological dualities and index theory on Carnot manifolds 22. Geometric K-homology with coefficients in elliptic complexes 23. Dualities on Carnot manifolds 24. Index theorems for H-elliptic operators 25. An outlook on graded Rockland sequences and their index theory Bibliography Theorem 3. Let X be an F F F-regular Carnot manifold. Then there exists a Hilbert space bundle H Ñ Γ X and a C 0 pΓ X q-linear˚-isomorphism πZ : I X Ñ C 0 pΓ X , KpHqqq. It holds that: i) The Hilbert space bundle H Ñ Γ X and the˚-isomorphism πZ is unique up to a line bundle on Γ X. ii) The inclusion I X ãÑ C˚pT H Xq induces a surjection in K-theory K˚pI X q Ñ K˚pC˚pT H Xqq. iii) There is a line bundle MpHq Ñ Γ X , uniquely determined up to isomorphism, such that for any other such H 1 and π 1 Z , there is an isomorphism H b MpHq-H 1 b MpH 1 q compatible with the pI X , C 0 pΓ X qq-bimodule structure and making the following diagram commutative
C*-algebra approach to the index theory of boundary value problems
Contemporary Mathematics, 2012
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 [5] showed how then the index can be computed in topological terms. The crucial observation is that elliptic operators can be mapped to compactly supported K-theory classes on the cotangent bundle over the interior of the manifold. The Atiyah-Singer topological index map, applied to this class, then furnishes the index of the operator. Based on this result, Fedosov, Rempel-Schulze and Grubb have given index formulas in terms of the symbols. In this paper we survey how C * -algebra K-theory, as initiated in , can be used to give a proof of Boutet de Monvel's index theorem for boundary value problems, a task carried out in , and how the same techniques yield an index theorem for families of Boutet de Monvel operators, detailed in . The key ingredient of our approach is a precise description of the K-theory of the kernel and of the image of the boundary symbol.
On Elliptic Differential Operators with Shifts II. The Cohomological Index Formula
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A cohomological formula for the Atiyah–Patodi–Singer index on manifolds with boundary
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The main result of this paper is a new Atiyah–Singer type cohomological formula for the index of Fredholm pseudodifferential operators on a manifold with boundary. The nonlocality of the chosen boundary condition prevents us to apply directly the methods used by Atiyah and Singer in [4, 5]. However, by using the K-theory of C*-algebras associated to some groupoids, which generalizes the classical K-theory of spaces, we are able to understand the computation of the APS index using classic algebraic topology methods (K-theory and cohomology). As in the classic case of Atiyah–Singer ([4, 5]), we use an embedding into a Euclidean space to express the index as the integral of a true form on a true space, the integral being over a C∞-manifold called the singular normal bundle associated to the embedding. Our formula is based on a K-theoretical Atiyah–Patodi–Singer theorem for manifolds with boundary that is inspired by Connes' tangent groupoid approach, it is not a groupoid interpreta...
Index Theory and Non-Commutative Geometry II. Dirac Operators and Index Bundles
Journal of K-Theory, 2007
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