Measures orthogonal to algebras and sets of antisymmetry

Abstract

IRVING GLICKSBERG(i) 1.We shall be concerned with two items concerning function algebras, connected in large part by their relation to the measures orthogonal to a given algebra : the expression of a general function algebra in terms of antisymmetric algebras recently obtained by Bishop [4], and the question of interpolation in such algebras.Let X he a compact Hausdorff space, C(X) the usual algebra of continuous complex functions on X, and A a closed subalgebra of C(X) containing the constants.ilov [16] called A antisymmetric if every real valued element of A is constant, and inquired whether (as a result of his [13, p. 127] partially suggested) every algebra of continuous functions could be expressed in terms of antisymmetric algebras.Recently Bishop [4] obtained the desired expression, following an approach suggested by Silov's work.In the first part of this paper we shall give a simpler approach to Bishop's theorem and some consequences.Call subset K of X a set of antisymmetry (or an antisymmetric set) of A if, for/in A, /real valued on K implies/is constant on K. Let f\K be the restriction of/to K, A\K = {f\K :feA}.Then we can restate Bishop's theorem as Theorem 1.1.Every antisymmetric set of A is contained in a maximal antisymmetric set.The collection Jf of maximal antisymmetric sets forms a pairwise disjoint, closed covering of X satisfying (a) feC(X) andf\KeA\Kfor every K in Jfimply J"e A;(h) A\K is closed in C(K), KeJT.The only nontrivial assertions are (a) and (b), of which the first generalizes the Stone-Weierstrass theorem.(When A is conjugate closed and separates the points of X, so that antisymmetric sets reduce to points, (a) is precisely Stone-Weierstrass.)Our proof of (a) is based on consideration of measures orthogonal to A and application of the Krein-Milman theorem, and is simply a modification of deBranges' proof of the Stone-Weierstrass theorem [6].In succeeding sections we consider some interpolation questions and their relation to measures orthogonal to A, as well as their relation to Jf.In 4 we generalize a result of Rudin [14] on norm preserving interpolation by continuous analytic functions on the disc to dirichlet algebras.