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Journal of Mathematical Sciences

, Volume 236, Issue 6, pp 641–662 | Cite as

Noncommutative Geometry and Analysis

  • A. G. SergeevEmail author
Article
  • 23 Downloads

Abstract

One of the main problems of noncommutative geometry is the translation of fundamental notions of analysis, topology, and differential geometry onto the language of Banach algebras. In this paper, we present a number of results of this kind focusing the attention on the noncommutative interpretation of the notions of differential and integral. Our presentation is based on the monographs Noncommutative Geometry by A. Connes and Elements of Noncommutative Geometry by J. M. Gracia-Bondia, J. C. Varilly, and H. Figueroa.

Keywords and phrases

C -algebra Dixmier trace Wodzicki residue differential graded algebra cycle Fredholm module Chern cocycle 

AMS Subject Classification

47L30 

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References

  1. 1.
    A. Connes, Noncommutative Geometry, Academic Press, London–San Diego (1994).Google Scholar
  2. 2.
    J. M. Gracia-Bondia, J. C. Varilly, and H. Figueroa, Elements of Noncommutative Geometry, Birkhäuser, Boston–Basel–Berlin (2001).Google Scholar
  3. 3.
    L. Hörmander, The Analysis of Linear Partial Differential Operators, Springer-Verlag, Berlin–Heidelberg (2003).Google Scholar
  4. 4.
    A. G. Sergeev, Lectures in Functional Analysis [in Russian], Steklov Mat. Inst., Moscow (2014).Google Scholar
  5. 5.
    M. E. Taylor, Pseudodifferential Operators, Princeton Math. Ser., 34, Princeton Univ. Press, Princeton, New Jersey (1981).Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Steklov Mathematical Institute of the Russian Academy of SciencesMoscowRussia

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