Approximation of continuous functions on pseudocompact spaces

Abstract

If ${\mathcal {S}^ * }$ is the family of subrings of ${C^ * }(X)$ such that if $S \in {\mathcal {S}^ * }$, $S$ contains the constant functions and is closed under uniform convergence, then the following are equivalent for a space $(X,\mathcal {J})$. (a) $(X,\mathcal {J})$ is pseudocompact. (b) If $S \in {\mathcal {S}^ * }$ functionally separates points and zero sets, $S$ generates $(X,\mathcal {J})$. (c) If $S \in {\mathcal {S}^ * }$ functionally separates zero sets, $S = {C^ * }(X)$. (d) If $S \in {\mathcal {S}^ * }$ generates the zero sets on $(X,\mathcal {J}),S = {C^ * }(X)$. (e) If $f \in S \in {\mathcal {S}^ * }$ and $Z(f) = \phi$ then $1/f \in S$ (even when it is required that $S$ generate the topology). (f) If $f \in S \in \mathcal {S}$ then $\left | f \right | \in S$ (even when it is required that $S$ generate the topology).