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Noncoincidence of the strict and strong operator topologies


Author: Joel H. Shapiro
Journal: Proc. Amer. Math. Soc. 35 (1972), 81-87
MSC: Primary 46E10
DOI: https://doi.org/10.1090/S0002-9939-1972-0306878-7
MathSciNet review: 0306878
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Abstract: Let E be an infinite-dimensional linear subspace of $ C(S)$, the space of bounded continuous functions on a locally compact Hausdorff space S. If $ \mu $ is a regular Borel measure on S, then each element of E may be regarded as a multiplication operator on $ {L^p}(\mu )(1 \leqq p < \infty )$. Our main result is that the strong operator topology this identification induces on E is properly weaker than the strict topology. For E the space of bounded analytic functions on a plane region G, and $ \mu $ Lebesgue measure on G, this answers negatively a question raised by Rubel and Shields in [9]. In addition, our methods provide information about the absolutely p-summing properties of the strict topology on subspaces of $ C(S)$, and the bounded weak star topology on conjugate Banach spaces.


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  • [1] R. C. Buck, Bounded continuous functions on a locally compact space, Michigan Math. J. 5 (1958), 95-104. MR 21 #4350. MR 0105611 (21:4350)
  • [2] J. B. Conway, The strict topology and compactness in the space of measures. II, Trans. Amer. Math. Soc. 126 (1967), 474-486. MR 34 #6503. MR 0206685 (34:6503)
  • [3] 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)
  • [4] P. R. Halmos, Measure theory, Van Nostrand, Princeton, N.J., 1950. MR 11, 504. MR 0033869 (11:504d)
  • [5] J. Kelley, I. Namioka et al., Linear topological spaces, University Series in Higher Math., Van Nostrand, Princeton, N.J., 1963. MR 29 #3851. MR 0166578 (29:3851)
  • [6] A. Lazar and J. Retheford, Nuclear spaces, Schauder bases, and Choquet Simplexes, Pacific J. Math. 37 (1971), 409-419. MR 0308730 (46:7844)
  • [7] A. Pietsch, Nukleare lokalkonvexe Räume, Schriftenreihe der Institute für Mathematik bei der Deutschen Akademie der Wissenschaften zu Berlin. Reihe A, Reine Mathematik, Heft 1, Akademie-Verlag, Berlin, 1965. MR 31 #6114. MR 0181888 (31:6114)
  • [8] -, Absolut p-summierende Abbildungen in normierten Räumen, Studia Math. 28 (1966/67), 333-353. MR 35 #7162. MR 0216328 (35:7162)
  • [9] L. A. Rubel and A. L. Shields, The space of bounded analytic functions on a region, Ann. Inst. Fourier (Grenoble) 16 (1966), fasc. 1, 235-277. MR 33 #6440. MR 0198281 (33:6440)
  • [10] J. Shapiro, Weak topologies on subspaces of $ C(S)$, Trans. Amer. Math. Soc. 157 (1971), 471-479. MR 0415285 (54:3375)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1972-0306878-7
Keywords: Strict topology, strong operator topology, absolutely p-summing topology, bounded continuous functions
Article copyright: © Copyright 1972 American Mathematical Society

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