Representation theorems for Banach algebras

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. ve 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–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.