Jensen measures without regularity

In this note we construct Swiss cheeses X such that R(X) is non-regular but such that R(X) has no non-trivial Jensen measures. We also construct a non-regular uniform algebra with compact, metrizable character space such that every point of the character space is a peak point. In [Co] Cole gave a counterexample to the peak point conjecture by constructing a non-trivial uniform algebra A with compact, metrizable character space Φ A such that every point of Φ A is a peak point for A. This uniform algebra was obtained from an example of McKissick [M] by a process of repeatedly adjoining square roots. Because McKissick's algebra is regular, so is this first example constructed by Cole (see [F2] and [Ka]). This leads to the following question: Let A be a uniform algebra with compact, metrizable character space Φ A such that every point of Φ A is a peak point for A. Must A be regular? In this note we construct an example of a Swiss cheese X for which the uniform algebra R(X) is non-regular, but such that R(X) has no non-trivial Jensen measures. We then apply Cole's construction to this example to produce an example of a non-regular uniform algebra with compact, metrizable character space such that every point of the character space is a peak point. We begin by recalling some standard facts about Jensen measures. Notation For a commutative Banach algebra A, we denote by Φ A the character space of A. Now suppose that A is a uniform algebra on a compact space X. For x ∈ X, let M x and J x be the ideals of functions in A vanishing at x, and in a neighbourhood of x, respectively. Definition 1 Let A be a uniform algebra on a compact space X, and let φ ∈ Φ A. Then a Jensen measure for φ is a regular, Borel probability measure µ on X such that, for all f ∈ A, log |φ(f)| ≤ X log |f (x)|dµ(x) (where log(0) is defined to be −∞). Let x ∈ X. We say that x is a point of continuity for A if there is no point y of X \ {x} satisfying M x ⊇ J y. It is elementary to see that if x is a point of continuity for A then the only Jensen measure for the evaluation character at x which is …