Elliptic Theory and Noncommutative Geometry: Nonlocal Elliptic Operators (Operator Theory: Advances and Applications Book 183)

The book deals with nonlocal elliptic differential operators. These are operators whose coefficients involve shifts generated by diffeomorphisms of the manifold on which the operators are defined. The main goal of the study is to relate analytical invariants (in particular, the index) of such operators to topological invariants of the manifold itself. This problem can be solved by modern methods of noncommutative geometry. To make the book self-contained, the authors have included necessary geometric material (C\*-algebras and their K-theory, cyclic homology, etc.). The Book Deals With Nonlocal Elliptic Differential Operators, Whose Coefficients Involve Shifts Generated By Diffeomorphisms Of The Manifold On Which The Operators Are Defined. The Main Goal Of The Study Is To Relate Analytical Invariants (in Particular, The Index) Of Such Operators To Topological Invariants Of The Manifold Itself. This Problem Can Be Solved By Modern Methods Of Noncommutative Geometry. To Make The Book Self-contained, The Authors Have Included Necessary Geometric Material (c[superscript*]-algebras And Their K-theory, Cyclic Homology, Etc.).--jacket. Analysis Of Nonlocal Elliptic Operators -- Nonlocal Functions And Bundles -- Nonlocal Elliptic Operators -- Elliptic Operators Over C*-algebras -- Homotopy Invariants Of Nonlocal Elliptic Operators -- Homotopy Classification -- Analytic Invariants -- Bott Periodicity -- Direct Image And Index Formulas In K-theory -- Chern Character -- Cohomological Index Formula -- Cohomological Formula For The ?-index -- Index Of Nonlocal Operators Over C*-algebras -- Examples -- Index Formula On The Noncommutative Torus -- An Application Of Higher Traces -- Index Formula For A Finite Group ?. Vladimir E. Nazaikinskii, Anton Yu. Savin, Boris Yu. Sternin. Includes Bibliographical References (p. [217]-221) And Index. This comprehensive yet concise book deals with nonlocal elliptic differential operators. These are operators whose coefficients involve shifts generated by diffeomorphisms of the manifold on which the operators are defined. This is the first book featuring a consistent application of methods of noncommutative geometry to the index problem in the theory of nonlocal elliptic operators. To make the book self-contained, the authors have included necessary geometric material.