One knows from the Gelfand
theory that every commutative semisimple Banach algebra A containing an
identity is a separating subalgebra of the algebra of all complex continuous
functions on the space of maximal ideals of A. We shall be concerned in
this paper with conditions which when imposed on a separating Banach
subalgebra A of C(Ω), Ω a compact Hausdorff space, will guarantee that
A = C(Ω). The conditions will take the form of restrictions on either the
algebra or the space Ω. For example we prove that if A is an 𝜖-normal Banach
subalgebra of C(Ω) then A = C(Ω) if an appropriate boundedness condition
holds locally on Ω. If Ω is assumed to be an F space in the sense of Gillman
and Henriksen this boundedness assumption is redundant. These results
include a recent characterization of Sidon sets in discrete groups due to
Rudin and have applications to interpolation problems for bounded analytic
functions.
