Let B denote a commutative,
semisimple Banach algebra with unit and let I be a fixed closed ideal in B. In the
maximal ideal space MB of B, fix a compact set X and put E = X ∩ h(I), where
h(I) is the hull of I. The main result of this note is the following
Theorem 1.1. Let I have an approximate unit that is uniformly bounded by the
constant C and let g be a nonnegative continuous function on X of sup-norm < 1
that vanishes on E. If h is an element of I and δ > 0, then there exists an element f
in I such that
