m at h . FA ] 1 4 Se p 19 98 STRONG REGULARITY FOR UNIFORM ALGEBRAS

A survey is given of the work on strong regularity for uniform algebras over the last thirty years, and some new results are proved, including the following. Let A be a uniform algebra on a compact space X and let E be the set of all those points x ∈ X such that A is not strongly regular at x. If E has no non-empty, perfect subsets then A is normal, and X is the character space of A. If X is either [0, 1] or the circle T and E is meagre with no non-empty, perfect subsets then A is trivial. These results extend Wilken's work from 1969. It is also shown that every separable Banach function algebra which has character space equal to either [0, 1] or T and has a countably-generated ideal lattice is uniformly dense in the algebra of all continuous functions. The study of strong regularity for uniform algebras was initiated by Wilken in 1969 [Wi2]. He had two main results. The first (a lemma) was that if a uniform algebra A is strongly regular on a space X then X is necessarily the character space of A, and A is normal. The second result was that there is no strongly regular uniform algebra on the unit interval [0, 1] other than the trivial one C[0, 1] itself. In this paper we survey the subsequent developments of the study, and extend Wilken's results. Various pertinent examples are given. We begin by introducing the terminology, definitions and notation which we shall need, along with some background results concerning strong regularity. Terminology and notation. In our terminology a compact space is a compact, Haus-dorff topological space. For any compact space X we denote the algebra of all continuous, complex-valued functions on X by C(X). Let A be a commutative, unital Banach algebra. We denote by Φ A the character space of A. Definition. Let X be a compact space. A function algebra on X is a subalgebra of C(X) which contains the constant functions and separates the points of X. A function algebra A on X is trivial if A = C(X). A Banach function algebra on X is a function algebra on X with a complete algebra norm. A uniform algebra on X is a Banach function algebra on X whose norm is the uniform norm on X.