S-Algebras on Sets in C n

Abstract

We give conditions which are necessary and sufficient for polynomial approximation of any continuous function on a compact subset of Cn. Let X be a compact set in Cn, complex n-space, P(X) the uniform closure of the polynomials on X, C(X) all continuous functions on X, m2n 2n-dimensional Lebesgue measure on Cn, and for any A in Cn let E(A)={z E Cn|Zi=Ai for some i}. A given set is a strong peak set if it is an intersection of peak sets and meets the boundary of each of them in a set which contains no nonempty perfect subsets. We say a Banach algebra A is an S-algebra if when x is in A and x, the Gelfand transform of x, vanishes at some p, then there exist xn in A such that xn vanish in (perhaps different) neighborhoods of p and j(xn-xI(--*0. For example, for any locally compact abelian group G, L'(G) is an S-algebra [6, p. 51]. The main question which motivates us here is: If A is a uniform algebra on a compact space X and A 'is an Salgebra, does A= C(X)? Our main result is the following. THEOREM. A necessary and sufficient condition that P(X)=C(X) is that (i) P(X) is an S-algebra, (ii) for almost all A E Cn with respect to m2', E(A) nX is a strong peak set, and (iii) each point of X is a peak point for P(X). We begin with some observations about uniform algebras which are S-algebras. LEMMA 1. Let A be a uniform algebra on a compact space X and suppose that A is an S-algebra. Then: (i) The maximal ideal space of A is X. (ii) A is normal. (iii) If each point of X is a peak point then A satisfies condition D [4, p. 86], i.e. iff E A and f (p)=O then there exist fn E A vanishing on neighborhoods of p such thatfInf-f. PROOF. (i) Let p be a homomorphism on A and y,u a representing measure for p with minimal closed support. If p, is not a point-mass then Presented to the Society, January 18, 1972 under the title A condition for polynomial approximation in Cn; received by the editors December 6, 1971 and, in revised form, October 18, 1972. AMS (MOS) subject classifications (1970). Primary 46J10; Secondary 41A10. ( American Mathematical Society 1973