C*-algebras and Elliptic Theory (original) (raw)

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