  For a space $X$ denote by $C_b(X)$ the Banach algebra of all continuous
bounded scalar-valued functions on $X$ and denote by $C_0(X)$ the set of all
elements in $C_b(X)$ which vanish at infinity.
  We prove that certain Banach subalgebras $H$ of $C_b(X)$ are isometrically
isomorphic to $C_0(Y)$, for some unique (up to homeomorphism) locally compact
Hausdorff space $Y$. The space $Y$ is explicitly constructed as a subspace of
the Stone--\v{C}ech compactification of $X$. The known construction of $Y$
enables us to examine certain properties of either $H$ or $Y$ and derive
results not expected to be deducible from the standard Gelfand Theory.
