On the Non-commutative Torus of Real Dimension Two

  • P. Cohen
Conference paper
Part of the Springer Proceedings in Physics book series (SPPHY, volume 47)

Abstract

The aim of the following notes is to recall how, according to the theory of Alain Connes, one may introduce a differentiable structure on a dense sub-algebra of the irrational rotation C*-algebra introduced in the lectures of Jean Bellissard. This sub-algebra may be viewed as a non-commutative generalisation of the classical torus of real dimension two. We compute the variation of the value at the origin of the meromorphic continuation of the zeta-function of the Laplacian on this algebra as the metric on the non-commutative torus varies within a given conformal class. This manifests the involved intervention into pseudo-differential computations of the non-commutativity, even for the simplest of non-commutative differential objects.

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References

  1. [1]
    T. Parker and S. Rosenberg, Invariants of Conformal Laplacians, Journal of Differential Geometry, vol. 25, 199–222, 1987.MathSciNetMATHGoogle Scholar
  2. [2]
    P. B. Gilkey, The Index Theorem and the Heat Equation, Publish or Perish, Boston, 1974.MATHGoogle Scholar
  3. [3]
    P. Cohen and A. Connes, in preparation.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1990

Authors and Affiliations

  • P. Cohen
    • 1
  1. 1.Laboratoire de Mathématiques Fondamentales, Equipe de Recherche Associée au CNRS 747, UER 48Université Pierre et Marie CurieParis Cedex 05France

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