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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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
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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DOI: http://dx.doi.org/10.1090/S0002-9939-1976-0420236-X
PII: S 0002-9939(1976)0420236-X
Article copyright: © Copyright 1976 American Mathematical Society