Elliptic Theory and Noncommutative Geometry: Nonlocal Elliptic Operators

Elliptic Theory and Noncommutative Geometry: Nonlocal Elliptic Operators


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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.

Product Details

ISBN-13: 9783764387747
Publisher: Birkhäuser Basel
Publication date: 06/23/2008
Series: Operator Theory: Advances and Applications , #183
Edition description: 2008
Pages: 224
Product dimensions: 6.60(w) x 9.20(h) x 0.70(d)

Table of Contents

Preface.- Introduction.- I. Analysis of Nonlocal Elliptic Operators.- 1. Nonlocal Functions and Bundles.- 2. Nonlocal Elliptic Operators.- 3. Elliptic Operators over C*-Algebras.- II. Homotopy Invariants of Nonlocal Elliptic Operators.- 5. Analytic Invariants.- 6. Bott Periodicity.- 7. Direct Image and Index Formulas in K-Theory.- 8. Chern Character.- 9. Cohomological Index Formula.- 10. Cohomological Formula for the Lambda-Index.- 11. Index of Nonlocal Operators over C*-Algebras.- III. Examples.- 12. Index Formula on the Noncommutative Torus.- 13. An Application of Higher Traces.- 14. Index Formula for a Finite Group Gamma.- IV. Appendices.- A. C*-Algebras.- A.1 Basic Notions.- A.2 Representations of C*-Algebras.- A.3 Tensor Products and Nuclear Algebras.- B. K-Theory of Operator Algebras.- B.1 Covariant K-Theory.- B.2 K-Homology.- C. Cyclic Homology and Cohomology.- C.1 Cyclic Cohomology.- C.2 Cyclic Homology.- Concise Bibliographical Remarks.- Bibliography.- Index

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