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On the Stone-Weierstrass theorem for the strict and superstrict topologies


Author: R. G. Haydon
Journal: Proc. Amer. Math. Soc. 59 (1976), 273-278
MSC: Primary 46E10
DOI: https://doi.org/10.1090/S0002-9939-1976-0420236-X
MathSciNet review: 0420236
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Abstract: Sentilles has introduced topologies $ {\beta _0},\;\beta $ and $ {\beta _1}$ on the space $ {C_b}(S)$ of all bounded, continuous, real-valued functions on the completely regular space $ S$, which yield as dual spaces the three important spaces of measures, $ {M_t}(S),\;{M_\tau }(S)$ and $ {M_\sigma }(S)$, respectively. A number of authors have proved a Stone-Weierstrass theorem for $ {\beta _0}$, the coarsest of the three topologies. In this paper, it is shown that the superstrict topology $ {\beta _1}$ does not obey the Stone-Weierstrass theorem, except perhaps when $ {\beta _1} = \beta $. Examples are then given to show that the situation for $ \beta $ itself is rather complicated.


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  • [1] G. Choquet, Lectures on analysis. Vol. I: Integration and topological vector spaces, Benjamin, New York, 1969. MR 40 #3252; erratum, 44, p. 1630. MR 0250011 (40:3252)
  • [2] D. H. Fremlin, D. J. H. Garling and R. G. Haydon, Bounded measures on topological spaces, Proc. London Math. Soc. (3) 25 (1972), 115-136. MR 0344405 (49:9144)
  • [3] R. Giles, A generalization of the strict topology, Trans. Amer. Math. Soc. 161 (1971), 467-474. MR 43 #7919. MR 0282206 (43:7919)
  • [4] J. D. Knowles, Measures on topological spaces, Proc. London Math. Soc. (3) 17 (1967), 139-156. MR 34 #4441. MR 0204602 (34:4441)
  • [5] C. Kuratowski, Topologie. Vol. I, 2nd ed., PWN, Warsaw, 1948; English transl., PWN, Warsaw; Academic Press, New York, 1966. MR 10, 389; 40 #840.
  • [6] E. A. Michael, $ {\aleph _0}$-spaces, J. Math. Mech. 15 (1966), 983-1002. MR 34 #6723. MR 0206907 (34:6723)
  • [7] F. D. Sentilles, Bounded continuous functions on a completely regular space, Trans. Amer. Math. Soc. 168 (1972), 311-336. MR 45 #4133. MR 0295065 (45:4133)
  • [8] W. H. Summers, The general complex bounded case of the strict weighted approximation problem, Math. Ann. 192 (1971), 90-98. MR 44 #2024. MR 0284800 (44:2024)
  • [9] -, Separability in the strict and substrict topologies, Proc. Amer. Math. Soc. 35 (1972), 507-514. MR 46 #2410. MR 0303272 (46:2410)
  • [10] F. Topsøe, Topology and measure, Springer-Verlag, Berlin, 1970. MR 0422560 (54:10546)
  • [11] A. C. M. van Rooij, Tight functionals and the strict topology, Kyungpook Math. J. 7 (1967), 41-43. MR 37 #3336. MR 0227752 (37:3336)
  • [12] V. S. Varadarajan, Measures on topological spaces, Mat. Sb. 55 (97) (1961), 35-100; English transl., Amer. Math. Soc. Transl. (2) 48 (1965), 161-228. MR 26 #6342. MR 0148838 (26:6342)

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DOI: https://doi.org/10.1090/S0002-9939-1976-0420236-X
Article copyright: © Copyright 1976 American Mathematical Society

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