Range transformations on a Banach function algebra. II

In this paper, localization for ultraseparability is introduced and a local version of Bernard's lemma is proven. By using these results it is shown that a function in Op(ID, Re A) is harmonic near the origin for a uniformly closed subalgebra A of CΌ(F) and an ideal I of A unless the uniform closure cl / of / is self-adjoint; in particular, it is shown that cl / is self-adjoint if Re / Re / c Re A9 which is not true when / is merely a closed subalgebra of A.