Let $X$ be a completely regular space . For a non-vanishing self-adjoint Banach subalgebra $H$ of $C_B(X)$ which has local units we construct the spectrum $\\mathfrak{sp}(H)$ of $H$ as an open suace of the Stone--\\v{C}ech compactification of $X$ which contains $X$ as a dense suace . The construction of $\\mathfrak{sp}(H)$ is simple . This enables us to study certain properties of $\\mathfrak{sp}(H)$ , 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 $\\mathfrak{sp}(H)$ is connected , locally connected and pseudocompact , strongly zero-dimensional , basically disconnected , extremally disconnected , or an $F$-space .