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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On the Stone-Weierstrass theorem for the strict and superstrict topologies
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by R. G. Haydon PDF
Proc. Amer. Math. Soc. 59 (1976), 273-278 Request permission

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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Additional Information
  • © Copyright 1976 American Mathematical Society
  • 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