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Arens multiplication and a characterization of $ w\sp{\ast} $-algebras


Author: T. W. Palmer
Journal: Proc. Amer. Math. Soc. 44 (1974), 81-87
MSC: Primary 46L10
DOI: https://doi.org/10.1090/S0002-9939-1974-0341122-8
MathSciNet review: 0341122
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Abstract: Let $ \mathfrak{A}$ be a Banach algebra which is the dual of a normed linear space $ \mathfrak{X}$. Suppose the multiplication in $ \mathfrak{A}$ is a continuous function of each factor separately in the weak* topology. We show that the natural projection of $ {\mathfrak{A}^{ \ast \ast }} = {\mathfrak{X}^{ \ast \ast \ast }}$ onto $ \mathfrak{A} = {\mathfrak{X}^ \ast }$ is a homomorphism with respect to either Arens' multiplication. From this we derive a simple proof of a variant form of Sakai's characterization of $ {W^ \ast }$-algebras.


References [Enhancements On Off] (What's this?)

  • [1] R. Arens, Operations induced in function classes, Monatsh. Math. 55 (1951), 1-19. MR 13, 372. MR 0044109 (13:372b)
  • [2] -, The adjoint of a bilinear operation, Proc. Amer. Math. Soc. 2 (1951), 839-848. MR 13, 659. MR 0045941 (13:659f)
  • [3] P. Civin and B. Yood, The second conjugate space of a Banach algebra as an algebra, Pacific J. Math. 11 (1961), 847-870. MR 26 #622. MR 0143056 (26:622)
  • [4] N. Dunford and J. T. Schwartz, Linear operators. I: General theory, Pure and Appl. Math., vol. 7, Interscience, New York, 1958. MR 22 #8302. MR 0117523 (22:8302)
  • [5] T. W. Palmer, General theory of $ ^ \ast $-algebras, Book in preparation.
  • [6] S. Sakai, A characterization of $ {W^ \ast }$-algebras, Pacific J. Math. 6 (1956), 763-773. MR 18, 811. MR 0084115 (18:811f)
  • [7] -, $ {C^ \ast }$-algebras and $ {W^ \ast }$-algebras, Springer, New York, 1971.
  • [8] J. Tomiyama, On the projection of norm one in $ {W^ \ast }$-algebras, Proc. Japan Acad. 33 (1957), 608-612. MR 20 #2635. MR 0096140 (20:2635)

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DOI: https://doi.org/10.1090/S0002-9939-1974-0341122-8
Article copyright: © Copyright 1974 American Mathematical Society

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