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