Measures Orthogonal to Algebras and Sets of Antisymmetry
Irving Glicksberg · Transactions of the American Mathematical Society · 1962
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.