  For a completely regular space $X$ and a non-vanishing self-adjoint closed
subalgebra $H$ of $C_B(X)$ which separates points from closed sets in $X$ we
construct the Gelfand spectrum $\mathfrak{sp}(H)$ of $H$ as an open subspace of
the compactification of $X$ generated by $H$. The simple construction of
$\mathfrak{sp}(H)$ enables easier examination of its properties. We illustrate
this by an example showing that the space $\mathfrak{sp}(H)$ is separable
metrizable if and only if $H$ is countably generated.
