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.
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Notes on Chapter VI
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.
H. Nakano, Hilbert algebras, Tôhoku Math. J., 2 (1950), 4–23.
W. Ambrose, The L2 -system of a unimodular group I, Trans. Amer. Math. Soc., 65 (1949), 27–48
R. Godement, Mémoire sur le théorie des caractères dans les groupes localement compacts unimodulaires, J. Math. Pure et Appl., 30 (1951), 1–110.
R. Godement, Théorie des caractères, I Algèbres unitaires, Ann. Math., 59 (1954), 47–62.
J. Dixmier, Algèbres quasi-unitaires, Comment. Math. Helv., 26 (1952), 275–322.
L. Pukanszky, On the theory of quasi-unitary algebras, Acta Sci. Math., 16 (1955), 103–121.
M. Tornita, Spectral theory of operator algebras. I, Math. J. Okayama Univ., 9 (1959), 63–98. II, ibid., 10 (1960), 19–60.
M. Takesaki, Tomita’s theory of modular Hilbert algebras and its applications, Lecture Notes in Math., Springer-Verlag, 128 (1970), 123.
I. E. Segal, An extension of Plancherel’c formula to separable unimodular groups, Ann. Math., 52 (1950), 272–292.
C. E. Sutherland, Direct integral theory for weights and the Plancherel formula, Bull. Amer. Math. Soc., 20 (1974), 456–461.
C. Lance, Direct integrals of left Hilbert algebras, Math. Ann., 216 (1975), 11–28.
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Takesaki, M. (2003). Left Hilbert Algebras. In: Theory of Operator Algebras II. Encyclopaedia of Mathematical Sciences, vol 125. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-10451-4_1
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DOI: https://doi.org/10.1007/978-3-662-10451-4_1
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