Extending continuous functions in zero-dimensional spaces

Abstract:

Suppose that $X$ is a completely regular, zero-dimensional space and that a dense subset $S$ of $X$ is not ${C^{\ast }}$-embedded in $X$; does there then exist a two-valued function in ${C^{\ast }}(S)$ with no continuous extension to $X$? The answer is negative even if $X$ is a compact space. The question was raised by N. J. Fine and L. Gillman in Extension of continuous functions in $\beta N$, Bull. Amer. Math. Soc. 66 (1960), 376-381.