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On $ W\sp{\ast} $ embedding of $ AW\sp{\ast} $-algebras


Author: Diane Laison
Journal: Proc. Amer. Math. Soc. 35 (1972), 499-502
MSC: Primary 46K99
DOI: https://doi.org/10.1090/S0002-9939-1972-0306928-8
MathSciNet review: 0306928
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Abstract: An $ A{W^ \ast }$-algebra N with a separating family of completely additive states and with a family $ \{ {e_\alpha }:\alpha \in A\} $ of mutually orthogonal projections such that $ {\operatorname{lub} _\alpha }{e_\alpha } = 1$ and $ {e_\alpha }N{e_\alpha }$ is a $ {W^ \ast }$-algebra for each $ \alpha \in A$ is shown to have a faithful representation as a ring of operators. This gives a new and considerably shorter proof that a semifinite $ A{W^ \ast }$-algebra with a separating family of completely additive states has a faithful representation as a ring of operators.


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

  • [1] J. Dixmier, Sur certains espaces considérés par M. H. Stone, Summa Brasil. Math. 2 (1951), 151-182. MR 14, 69. MR 0048787 (14:69e)
  • [2] J. Feldman, Embedding of $ A{W^\ast}$-algebras, Duke Math. J. 23 (1956), 303-307. MR 17, 1229. MR 0078669 (17:1229a)
  • [3] I. Kaplansky, Projections in Banach algebras, Ann. of Math. (2) 53 (1951), 235-249. MR 13, 48. MR 0042067 (13:48b)
  • [4] K. Saitô, Non-commutative extension of Lusin's theorem, Tôhoku Math. J. 19 (1967), 332-340. MR 37 #2007. MR 0226417 (37:2007)
  • [5] -, A non-commutative theory of integration for a semi-finite $ A{W^\ast}$-algebra and a problem of Feldman, Tôhoku Math. J. 22 (1970), 420-461. MR 0275182 (43:939)
  • [6] S. Sakai, $ {C^ \ast }$-algebras and $ {W^ \ast }$-algebras, Springer-Verlag, New York, 1971. MR 0442701 (56:1082)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1972-0306928-8
Article copyright: © Copyright 1972 American Mathematical Society

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