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Elliptic Theory and Noncommutative Geometry

Nonlocal Elliptic Operators

  • Book
  • © 2008

Overview

Authors:
  1. Vladimir E. Nazaikinskii

    1. Institute for Problems in Mechanics, Russian Academy of Sciences, Moscow, Russia

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  2. Anton Yu. Savin

    1. Independent University of Moscow, Moscow, Russia

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  3. Boris Yu. Sternin

    1. Independent University of Moscow, Moscow, Russia

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  • First and so far the only book featuring a consistent application of methods of noncommutative geometry to the index problem in the theory of nonlocal elliptic operators
  • Provides important results, which are in a sense definitive, on the above-mentioned topic and at the same time contains all the necessary preliminary material, thus being effectively self-contained
  • Not only comprehensive but also rather concise, which makes it even more of an attraction for readers at any level
  • Can serve as a concise introduction to noncommutative geometry, which is one of the most powerful tools for studying a wide range of problems in mathematics and theoretical physics
  • Includes supplementary material: sn.pub/extras

Part of the book series: Operator Theory: Advances and Applications (OT, volume 183)

Part of the book sub series: Advances in Partial Differential Equations (APDE)

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Table of contents (15 chapters)

  1. Front Matter

    Pages i-xii
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  2. Introduction

    1. Introduction

      Pages 1-2

  3. Analysis of Nonlocal Elliptic Operators

    1. Front Matter

      Pages 3-3
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    2. Nonlocal Functions and Bundles

      Pages 5-25

    3. Nonlocal Elliptic Operators

      Pages 27-36

    4. Elliptic Operators over C*-Algebras

      Pages 37-55

  4. Homotopy Invariants of Nonlocal Elliptic Operators

    1. Front Matter

      Pages 57-57
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    2. Homotopy Classification

      Pages 59-62

    3. Analytic Invariants

      Pages 63-65

    4. Bott Periodicity

      Pages 67-74

    5. Direct Image and Index Formulas in K-Theory

      Pages 75-87

    6. Chern Character

      Pages 89-98

    7. Cohomological Index Formula

      Pages 99-114

    8. Cohomological Formula for the Λ-Index

      Pages 115-125

    9. Index of Nonlocal Operators over C*-Algebras

      Pages 127-136

  6. Back Matter

    Pages 157-224
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Keywords

About this book

Noncommutative geometry, which can rightfully claim the role of a philosophy in mathematicalstudies,undertakesto replacegoodoldnotionsofclassicalgeometry (suchas manifolds,vectorbundles, metrics, di?erentiable structures,etc. ) by their abstract operator-algebraic analogs and then to study the latter by methods of the theory of operator algebras. At ?rst sight, this pursuit of maximum possible generality harbors the danger of completely forgetting the classical beginnings, so that not only the answers but also the questions would defy stating in traditional terms. Noncommutative geometry itself would become not only a method but also the main subject of investigation according to the capacious but not too practical formula: “Know thyself. ” Fortunately, this is not completely true (or even is completely untrue) in reality: there are numerous problems that are quite classical in their statement (or at least admit an equivalent classical statement) but can be solved only in the framework of noncommutative geometry. One of such problems is the subject of the present book. The classical elliptic theory developed in the well-known work of Atiyah and Singer on the index problem relates an analytic invariant of an elliptic pseud- i?erential operator on a smooth compact manifold, namely, its index, to topol- ical invariants of the manifold itself. The index problem for nonlocal (and hence nonpseudodi?erential) elliptic operators is much more complicated and requires the use of substantially more powerful methods than those used in the classical case.

Authors and Affiliations

  • Institute for Problems in Mechanics, Russian Academy of Sciences, Moscow, Russia

    Vladimir E. Nazaikinskii

  • Independent University of Moscow, Moscow, Russia

    Anton Yu. Savin, Boris Yu. Sternin

Bibliographic Information

  • Book Title: Elliptic Theory and Noncommutative Geometry

  • Book Subtitle: Nonlocal Elliptic Operators

  • Authors: Vladimir E. Nazaikinskii, Anton Yu. Savin, Boris Yu. Sternin

  • Series Title: Operator Theory: Advances and Applications

  • DOI: https://doi.org/10.1007/978-3-7643-8775-4

  • Publisher: Birkhäuser Basel

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: Birkhäuser Basel 2008

  • Hardcover ISBN: 978-3-7643-8774-7Published: 16 May 2008

  • eBook ISBN: 978-3-7643-8775-4Published: 30 June 2008

  • Series ISSN: 0255-0156

  • Series E-ISSN: 2296-4878

  • Edition Number: 1

  • Number of Pages: XII, 224

  • Topics: Operator Theory

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