Let A be a function algebra
with its maximal ideal space MA. Let B be a function algebra such that
A ⊂ B ⊂ C(MA). What can be said about MB? We prove that MA= MB if every
point x ∈ MA has a fundamental neighborhood system {W} such that the
topological boundary bW of each W is contained in the Choquet boundary
of A or if A is a normal function algebra. The first condition is satisfied
if MA is a one dimensional topological space. Let H(A) be the function
algebra on MA generated by all functions which are locally approximable in
A. We prove that MH(A)= MA and then we try to generalize this result.
If f ∈ C(MA) is such that f is locally approximable in A at every point
where f is different from zero then MA is the maximal ideal space of the
function algebra generated by A and f. We also look at closed subsets F of
MA such that MH(F)= F where H(F) is the function algebra generated
by restricting to F all functions that are defined and locally approximable
in A in some neighborhood of F. These sets are called natural sets. We
prove that there exists a smallest natural set B(F) containing a closed set
F in MA and that the Silov boundary of H(B(F)) is contained in F. We
also find conditions that guarantee that a closed set in MA is a natural
set.
