Existence of maximal ideals in algebras of continuous functions

Abstract

This note is a conitribution to the study of the subalgebras of the space C(S) of complex-valued continuous functions on a compact Hausdorff space S. We are interested here in finding conditions under which an algebra has maximal ideals other than the obvious ones corresponding to the points of S. We shall restrict ourselves to the case where S is a circle or an interval and shall give two sets of hypotheses under which other maximal ideals do exist. Both theorems depend on deep results about algebras of functions. The first is really a corollary of the theorem of Mergelyan and others which states that oIn any compact set E in the plane, of plane measure zero, an arbitrary continuous function can be uniformly approximated by rational functions having their poles outside E. We are presenting Theorem 1 mainly because its corollary is a statement about polynomials in several complex variables which seems to be new, and which we think is curious. Theorem 2 depends less obviously on the theorem of Silov asserting the existence of idempotents corresponding to the open-closed subsets of the structure space of a commutative Banach algebra. Our theorem is of Stone-Weierstrass type; it states, under hypotheses, that a given algebra either has many maximal ideals or else contains all continuous functions. Let [ be a closed subalgebra of C(S), and let 9 be the collection of functions in W which vanish at a given point of S. Either M is all of 2f, or 9 is a maximal ideal in W, in which case we say that 9J is associated with the given point. If 9 is a maximal ideal which is not of this form, we say that it is not associated with any point of S. We can now state our first theorem.