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Left Hilbert Algebras

Chapter
Part of the Encyclopaedia of Mathematical Sciences book series (EMS, volume 125)

Abstract

This chapter is devoted to the foundation of non-commutative integration theory. In the first volume of this book, we have seen the strong similarity between the theory of operator algebras and the integration theory. To explore this similarity further, it is necessary to work on the theory of left Hilbert algebras. It is the non-commutative counter part of the algebra of all bounded square integrable functions on a measure space.

Keywords

Measurable Field Operator Algebra Closed Operator Polar Decomposition Graph Norm 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Notes on Chapter VI

  1. [430]
    H. Araki and E. J. Woods, Representations of the canonical commutation relations describing a non-relativistic infinite free Bose gas, J. Math. Phys., 4 (1963), 637–662.MathSciNetCrossRefGoogle Scholar
  2. [646]
    H. Nakano, Hilbert algebras, Tôhoku Math. J., 2 (1950), 4–23.zbMATHCrossRefGoogle Scholar
  3. [420]
    W. Ambrose, The L2 -system of a unimodular group I, Trans. Amer. Math. Soc., 65 (1949), 27–48MathSciNetzbMATHGoogle Scholar
  4. [528]
    R. Godement, Mémoire sur le théorie des caractères dans les groupes localement compacts unimodulaires, J. Math. Pure et Appl., 30 (1951), 1–110.MathSciNetGoogle Scholar
  5. [529]
    R. Godement, Théorie des caractères, I Algèbres unitaires, Ann. Math., 59 (1954), 47–62.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [502]
    J. Dixmier, Algèbres quasi-unitaires, Comment. Math. Helv., 26 (1952), 275–322.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [672]
    L. Pukanszky, On the theory of quasi-unitary algebras, Acta Sci. Math., 16 (1955), 103–121.MathSciNetzbMATHGoogle Scholar
  8. [720]
    M. Tornita, Spectral theory of operator algebras. I, Math. J. Okayama Univ., 9 (1959), 63–98. II, ibid., 10 (1960), 19–60.Google Scholar
  9. [708]
    M. Takesaki, Tomita’s theory of modular Hilbert algebras and its applications, Lecture Notes in Math., Springer-Verlag, 128 (1970), 123.MathSciNetGoogle Scholar
  10. [689]
    I. E. Segal, An extension of Plancherel’c formula to separable unimodular groups, Ann. Math., 52 (1950), 272–292.CrossRefGoogle Scholar
  11. [696]
    C. E. Sutherland, Direct integral theory for weights and the Plancherel formula, Bull. Amer. Math. Soc., 20 (1974), 456–461.MathSciNetCrossRefGoogle Scholar
  12. [623]
    C. Lance, Direct integrals of left Hilbert algebras, Math. Ann., 216 (1975), 11–28.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of CaliforniaLos AngelesUSA

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