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Correspondences and Index

We define a certain class of correspondences of polarized representations of C*-algebras. Our correspondences are modeled on the spaces of boundary values of elliptic operators on bordisms joining two manifolds. In this setup we define the index. The main subject of the paper is the additivity of the index.
Bogdan Bojarski, Andrzej Weber

Approximation Properties for Discrete Groups

We provide an illustration of an interesting and nontrivial interaction between analytic and geometric properties of a group. We provide a short survey of approximation properties of operator algebras associated with discrete groups. We then demonstrate directly that groups that satisfy the property RD with respect to a conditionally negative length function have the metric approximation property.
Jacek Brodzki, Graham A. Niblo

A Riemannian Invariant, Euler Structures and Some Topological Applications

First we discuss 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 Chern-Simons class from a pair of two Riemannian metrics to a pair of a Riemannian metric and a smooth triangulation. Next we discuss a generalization of Turaev’s Euler structures to manifolds with non-vanishing Euler characteristics and introduce the Poincarée 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 the Ray-Singer torsion. The Bismut-Zhang theorem can then be reformulated as a statement comparing two genuine topological invariants.
Dan Burghelea, Stefan Haller

Morse Inequalities for Foliations

We outline the analytical proof of the Morse inequalities for measured foliations obtained in [2] and give some applications. The proof is based on the use of a twisted Laplacian.
Alain Connes, Thierry Fack

Index Theory for Generalized Dirac Operators on Open Manifolds

In the first part of the paper, we give a short review of index theory on open manifolds. In the second part, we establish a general relative index theorem admitting compact topological perturbations and Sobolev perturbations of all other ingredients.
Jürgen Eichhorn

Semiclassical Asymptotics and Spectral Gaps for Periodic Magnetic Schrödinger Operators on Covering Manifolds

We survey a method to prove the existence of gaps in the spectrum of periodic second-order elliptic partial differential operators, which was suggested by Kordyukov, Mathai and Shubin, and describe applications of this method to periodic magnetic Schrödinger operators on a Riemannian manifold, which is the universal covering of a compact manifold. We prove the existence of arbitrarily large number of gaps in the spectrum of these operators in the asymptotic limits of the strong electric field or the strong magnetic field under Morse type assumptions on the electromagnetic potential. We work on the level of spectral projections (and not just their traces) and obtain an asymptotic information about classes of these projections in K-theory. An important corollary is a vanishing theorem for the higher traces in cyclic cohomology for the spectral projections. This result is then applied to the quantum Hall effect.
Yuri A. Kordyukov

The Group of Unital C*-extensions

Let A and B be separable C*-algebras, A unital and B stable. It is shown that there is a natural six-terms exact sequence which relates the group which arises by considering all semi-split extensions of A by B to the group which arises by restricting the attention to unital semi-split extensions of A by B. The six-terms exact sequence is an unpublished result of G. Skandalis.
Vladimir Manuilov, Klaus Thomsen

Lefschetz Theory on Manifolds with Singularities

The semiclassical method in Lefschetz theory is presented and applied to the computation of Lefschetz numbers of endomorphisms of elliptic complexes on manifolds with singularities. Two distinct cases are considered, one in which the endomorphism is geometric and the other in which the endomorphism is specified by Fourier integral operators associated with a canonical transformation. In the latter case, the problem includes a small parameter and the formulas are (semiclassically) asymptotic. In the first case, the parameter is introduced artificially and the semiclassical method gives exact answers. In both cases, the Lefschetz number is the sum of contributions of interior fixed points given (in the case of geometric endomorphisms) by standard formulas plus the contribution of fixed singular points. The latter is expressed as a sum of residues in the lower or upper half-plane of a meromorphic operator expression constructed from the conormal symbols of the operators involved in the problem.
Vladimir Nazaikinskii, Boris Sternin

Residues and Index for Bisingular Operators

We consider an algebra of pseudo-differential operators on the product of two manifolds which contains, in particular, the tensor products of usual pseudo-differential operators. For that algebra we discuss the existence of trace functionals like Wodzicki’s residue and we prove a homological index formula for the elliptic elements.
Fabio Nicola, Luigi Rodino

On the Hopf-type Cyclic Cohomology with Coefficients

In this note we discuss the Hopf-type cyclic cohomology with coefficients, introduced in the paper [1]: we calculate it in a couple of interesting examples and propose a general construction of coupling between algebraic and coalgebraic version of such cohomology, taking values in the usual cyclic cohomology of an algebra.
I. M. Nikonov, G. I. Sharygin

The Thom Isomorphism in Gauge-equivariant K-theory

In a previous paper [14], we have introduced the gauge-equivariant K-theory group \( K_\mathcal{G}^0 (X)\) of a bundle πX : X → B endowed with a continuous action of a bundle of compact Lie groups \( p:\mathcal{G} \to B\). These groups are the natural range for the analytic index of a family of gauge-invariant elliptic operators (i.e., a family of elliptic operators invariant with respect to the action of a bundle of compact groups). In this paper, we continue our study of gauge-equivariant K-theory. In particular, we introduce and study products, which helps us establish the Thom isomorphism in gauge-equivariant K-theory. Then we construct push-forward maps and define the topological index of a gauge-invariant family.
Victor Nistor, Evgenij Troitsky

Pseudodifferential Subspaces and Their Applications in Elliptic Theory

The aim of this paper is to explain the notion of subspace defined by means of pseudodifferential projection and give its applications in elliptic theory. Such subspaces are indispensable in the theory of well-posed boundary value problems for an arbitrary elliptic operator, including the Dirac operator, which has no classical boundary value problems. Pseudodifferential subspaces can be used to compute the fractional part of the spectral Atiyah73-Patodi— Singer eta invariant, when it defines a homotopy invariant (Gilkey’s problem). Finally, we explain how pseudodifferential subspaces can be used to give an analytic realization of the topological K-group with finite coefficients in terms of elliptic operators. It turns out that all three applications are based on a theory of elliptic operators on closed manifolds acting in subspaces.
Anton Savin, Boris Sternin

New L 2-invariants of Chain Complexes and Applications

We study the homotopy invariants of free cochain complexes and Hilbert complex. This invariants are applied to calculation of exact values of Morse numbers of smooth manifolds.
Vladimir V. Sharko

Bundles of C*-algebras and the KK(X;−,−)-bifunctor

An overview about C*-algebra bundles with a ℤ-grading is presented, with particular emphasis on classification questions. In particular, we discuss the role of the representable KK(X;−,−)-bifunctor introduced by Kasparov. As an application, we consider Cuntz-Pimsner algebras associated with vector bundles, and give a classification in terms of K-theoretical invariants in the case in which the base space is an n-sphere.
Ezio Vasselli
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