Communications in Mathematical Physics

, Volume 2, Issue 1, pp 31–48 | Cite as

C*-Algèbres des systèmes canoniques. I

  • G. Loupias
  • S. Miracle-Sole


The “twisted convolution” associated with the Weyl form of the canonical commutation relations forn degrees of freedom is decribed using ordinary convolution on a nilpotent central extension of additive phase space by the one-dimensional torus. Twisted convolution determines severalC*-algebras of quantum mechanical observables amongst which we study especially the algebra ℒ2(\(\mathfrak{E}\), σ) consisting of the ℒ2-functions on phase space and mapped isometrically onto the Hilbert-Schmidt-operators by the Schrödinger representation. The two last sections of the paper deal with “phase space quantum mechanics” from the point of view of twisted convolution: theWigner-Moyal formalism and the entire function formalism ofBargmann andSegal.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Kastler, D.: Commun. math. Phys.1, 14 (1965).Google Scholar
  2. [2]
    Mackey, G. W.: Acta Math.99, 265 (1958).Google Scholar
  3. [3]
    Wigner, E.: Phys. Rev.40, 749 (1932).Google Scholar
  4. [3a]
    Moyal, J. E.: Proc. Cambridge Phil. Soc.45, 99 (1949).Google Scholar
  5. [4]
    Baker, G. A.: Phys. Rev.109, 2198 (1958).Google Scholar
  6. [5]
    Bargmann, V.: Commun. Pure and Appl. Math.14, 187 (1961).Google Scholar
  7. [5a]
    Segal, I.: Illinois J. Math.6, 500 (1962).Google Scholar
  8. [6]
    Michel, L.: Invariance in Quantum Mechanics and Group extensions. Manuscript, I.H.E.S. Bures-sur-Yvette. Seine et Oise, France.Google Scholar
  9. [7]
    Chevalley, C.: Théorie des groupes de Lie. Tome III. Actualités Scientifiques et Industrielles. Paris: Hermann 1955.Google Scholar
  10. [8]
    Dixmier, J.: LesC*-alèbres et leurs représentation. Paris: Gauthier-Villars 1964.Google Scholar
  11. [9]
    Fell, J. M. G.: Proc. Am. Math. Soc.13, 93 (1962).Google Scholar
  12. [10]
    Hewitt, E., andK. A. Ross: Abstract Harmonic Analysis. Berlin-Göttingen-Heidelberg: Springer 1963.Google Scholar
  13. [11]
    Rudin, W.: Fourier Analysis on Groups. New York: John Wiley 1962.Google Scholar
  14. [12]
    Dixmier, J.: Les Algèbres d'opérateurs dans l'Espace Hilbertien. Paris: Gauthier-Villars 1957.Google Scholar
  15. [13]
    Weyl, H.: The Theory of Groups and Quantum Mechanics. London: Methnen 1931.Google Scholar
  16. [14]
    Bourbaki, N.: Espaces vectoriels topologiques. Fascicule de Résultats. Actualités Scientifiques et Industrielles. Paris: Hermann 1955.Google Scholar
  17. [15]
    Rickart, C. E.: General Theory of Banach Algebras. Princeton-London-Toronto: Van Nostrand (1960).Google Scholar
  18. [16]
    Schwartz, L.: Théorie des Distributions. Tome II. Actualités Scientifiques et Industrielles. Paris: Hermann 1959.Google Scholar
  19. [17]
    Souriau, J. M.: Commun. math. Phys1, 374–398 (1966).Google Scholar

Copyright information

© Springer-Verlag 1966

Authors and Affiliations

  • G. Loupias
    • 1
  • S. Miracle-Sole
    • 1
  1. 1.Physique ThéoriqueUniversité D'Aix-MarseilleFrance

Personalised recommendations