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On Boolean algebras of projections and scalar-type spectral operators


Author: W. Ricker
Journal: Proc. Amer. Math. Soc. 87 (1983), 73-77
MSC: Primary 47B40; Secondary 47D30
DOI: https://doi.org/10.1090/S0002-9939-1983-0677235-6
MathSciNet review: 677235
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Abstract: It is shown that the weakly closed operator algebra generated by an equicontinuous $ \sigma $-complete Boolean algebra of projections on a quasi-complete locally convex space consists entirely of scalar-type operators. This extends W. Badé's well-known theorem that the same assertion is valid for Banach spaces; however, the technique of proof here differs from his method, which extends only to metrizable spaces.


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

  • [1] W. G. Badé, On Boolean algebras of projections and algebras of operators, Trans. Amer. Math. Soc. 80 (1955), 345-359. MR 0073954 (17:513d)
  • [2] N. Dunford and J. T. Schwartz, Linear operators, vol. III, Interscience, New York, 1971.
  • [3] I. Kluvánek and G. Knowles, Vector measures and control systems, North-Holland, Amsterdam, 1976.
  • [4] B. Walsh, Structure of spectral measures on locally convex spaces, Trans. Amer. Math. Soc. 120 (1965), 295-326. MR 0196503 (33:4690)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1983-0677235-6
Keywords: Closed measure, spectral measure, scalar-type spectral operator, Boolean algebra of projections
Article copyright: © Copyright 1983 American Mathematical Society

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