Index of elliptic operators for a diffeomorphism

We develop elliptic theory of operators associated with a diffeomorphism of a closed smooth manifold. The aim of the present paper is to obtain an index formula for such operators in terms of topological invariants of the manifold and of the symbol of the operator. The symbol in this situation is an element of a certain crossed product. We express the index as the pairing of the class in K-theory defined by the symbol and the Todd class in periodic cyclic cohomology of the crossed product.Comment: 37 page

Elliptic theory for operators associated with diffeomorphisms of smooth manifolds

In this paper we give a survey of elliptic theory for operators associated with diffeomorphisms of smooth manifolds. Such operators appear naturally in analysis, geometry and mathematical physics. We survey classical results as well as results obtained recently. The paper consists of an introduction and three sections. In the introduction we give a general overview of the area of research. For the reader's convenience here we tried to keep special terminology to a minimum. In the remaining sections we give detailed formulations of the most important results mentioned in the introduction.Comment: 27 pages, 2 figure

On the homotopy classification of elliptic operators on stratified manifolds

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.Comment: 28 pages; submitted to Izvestiya Ross. Akad. Nau

Guillemin Transform and Toeplitz Representations for Operators on Singular Manifolds

An approach to the construction of index formulas for elliptic operators on singular manifolds is suggested on the basis of K-theory of algebras and cyclic cohomology. The equivalence of Toeplitz and pseudodifferential quantizations, well known in the case of smooth closed manifolds, is extended to the case of manifolds with conical singularities. We describe a general construction that permits one, for a given Toeplitz quantization of a C^*-algebra, to obtain a new equivalent Toeplitz quantization provided that a resolution of the projection determining the original quantization is given.Comment: 26 page