Let A be a function algebra
with its maximal ideal space M_{A}. Let B be a function algebra such that
A ⊂ B ⊂ C(M_{A}). What can be said about M_{B}? We prove that M_{A} = M_{B} if every
point x ∈ M_{A} 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 M_{A} is a one dimensional topological space. Let H(A) be the function
algebra on M_{A} generated by all functions which are locally approximable in
A. We prove that M_{H(A)} = M_{A} and then we try to generalize this result.
If f ∈ C(M_{A}) is such that f is locally approximable in A at every point
where f is different from zero then M_{A} is the maximal ideal space of the
function algebra generated by A and f. We also look at closed subsets F of
M_{A} such that M_{H(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 M_{A} and that the Silov boundary of H(B(F)) is contained in F. We
also find conditions that guarantee that a closed set in M_{A} is a natural
set.
