Representation theorems for Banach algebras

For a space X   denote by Cb(X)Cb(X) the Banach algebra of all continuous bounded scalar-valued functions on X   and denote by C0(X)C0(X) the set of all elements in Cb(X)Cb(X) which vanish at infinity.We prove that certain Banach subalgebras H   of Cb(X)Cb(X) are isometrically isomorphic to C0(Y)C0(Y), for some unique (up to homeomorphism) locally compact Hausdorff space Y. The space Y is explicitly constructed as a subspace of the Stone–Čech 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.