Vol. 39, No. 3, 1971

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Restrictions of Banach function spaces

Donald Richard Chalice

Vol. 39 (1971), No. 3, 593–602

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.

Mathematical Subject Classification 2000
Primary: 46J10
Received: 29 April 1970
Published: 1 December 1971
Donald Richard Chalice