A Gelfand-Naimark type theorem

Let X be a completely regular space. For a non-vanishing self-adjoint Banach subalgebra H of CB(X) which has local units we construct the spectrum sp(H) of H as an open subspace of the Stone-Čech compactification of X which contains X as a dense subspace. The construction of sp(H) is simple. This enables us to study certain properties of sp(H), among them are various compactness and connectedness properties. In particular, we find necessary and sufficient conditions in terms of either H or X under which sp(H) is connected, locally connected and pseudocompact, strongly zero-dimensional, basically disconnected, extremally disconnected, or an F-space.