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

Haydon, R. G.

doi:10.1090/S0002-9939-1976-0420236-X

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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