Slices in the Unit Ball of a Uniform Algebra

We show that every nonvoid relatively weakly open subset, in particular every slice, of the unit ball of an infinite-dimensional uniform algebra has diameter 2. It is an important task in Banach space theory to determine the extreme point structure of the unit ball for various examples of Banach spaces. The most common way to describe “corners” of convex sets is by looking for extreme points, exposed points, denting points and strongly exposed points. Every strongly exposed point is both denting and exposed and every denting or exposed point is extreme. In this note A denotes an infinite-dimensional uniform algebra, i.e., an infinite-dimensional closed subalgebra of some C(K)-space which separates the points of K and contains the constant functions. In [1] Beneker and Wiegerinck demonstrated the non-existence of strongly exposed points in BA, the closed unit ball of A. Here we shall prove a stronger result by more elementary means. A corollary of our result is that the set of denting points is, in fact, also empty. Recall that we may assume that K is the Silov boundary of A. It is a fundamental result in the theory of uniform algebras that then the set of strong boundary points is dense in K; cf. [4, p. 48 and p. 78]. (A point x ∈ K is a strong boundary point if for every neighbourhood V of x and every δ > 0 there is some f ∈ A such that f(x) = ‖f‖ = 1 and |f | ≤ δ off V .) We now turn to our first result, which gives a quantitative statement of the non-dentability of BA. Theorem 1. Every slice of the unit ball of an infinite-dimensional uniform algebra A has diameter 2. Proof. Take an arbitrary slice S = {a ∈ BA: re l(a) ≥ 1−ε}, where ‖l‖ = 1. We will produce two functions in S having distance nearly 2. Let 0 < δ ≤ ε/11. We first pick some f ∈ BA such that re l(f) ≥ 1− δ. The functional l can be represented by a regular Borel measure μ on K with ‖μ‖ = 1, i.e., l(a) = ∫ K a dμ for all a ∈ A. Let ∅ 6= V0 ⊂ K be an open set with |μ|(V0) ≤ δ; such a set exists since K is infinite. Fix a strong boundary point x0 ∈ V0. Using the definition of a strong boundary point, inductively construct functions g1, g2, . . . ∈ A and nonvoid open subsets Date: February 1, 2008. 1991 Mathematics Subject Classification. Primary: 46B20.