Let X be a compact Hausdorfs
space. Let C(X) be the space of continuous complex-valued functions on X and A be
a function algebra on X, that is a uniformly closed separating subalgebra of C(X)
containing the constants. If F is a closed subset of X we say that A interpolates on F
if A|F = C(F). By a positive measure μ we shall always mean a positive regular
bounded Borel measure on X. Let F be a measurable subset of X. We say a subspace
S of Lp(μ) interpolates on F if S|F = Lp(F) = Lp(μF), where μF is the restriction of
μ to F. Let HP(μ) be the closure of A in Lp(μ) where 1 ≦ p < ∞, and let
H∞(μ) = H2(μ) ∩ L∞(μ). One question we are concerned with here is whether
interpolation of the algebra is sufficient to imply interpolation of its associated
Hp-spaces. We therefore begin by obtaining necessary and sufficient conditions for a
closed subspace of Lp(μ) to have closed restriction in Lp(F). These condition
are analogous to some obtained by Glicksberg for function algebras. Using
these results we obtain theorems about interpolation of certain invariant
subspaces, and then apply them to Hp-spaces. In particular we show that when A
approximates in modulus and μ is any measure which is not a point-mass, Hp(μ)
interpolates only on sets of measure zero. (One sees that A interpolates
only on sets of measure zero, so our original question has a trivial answer
for these algebras.) For uniformly closed weak-star Dirichlet algebras again
the answer to our original question is affirmative. Finally we provide an
example of an algebra which interpolates such that H∞(μ) interpolates and
the Hp(μ) do not interpolate for 1 ≦ p < ∞. I am indebted to a paper of
Glicksberg for those techniques which inspired the present effort. Below we
show that these techniques apply to the Lp situation and to other “similar”
situations.
