Representing measures for the disc algebra and for the ball algebra

Abstract

We consider the set of representing measures at 0 for the disc and the ball algebra. The structure of the extreme elements of these sets is investigated. We give particular attention to representing measures for the 2-ball algebra which arise by lifting representing measures for the disc algebra. Introduction. Let X be a compact Hausdorff space, C(X) the algebra of continuous functions on X with the supremum norm, A a function algebra on X and Φ a multiplicative linear functional on A. The set of all probability measures on X which represent Φ is denoted by MΦ. As is well known, MΦ is a nonempty, weak∗-compact, convex subset of the set of all regular Borel measures, M(X), which is viewed as the dual space of C(X). For a compact X ⊂ C, the function algebra P (X) is defined as the closure of the holomorphic polynomials in C(X). Let B = B be the unit ball in C, n ≥ 2, S its boundary. Next, D is the open unit disc in C, D its closure and T its boundary. Traditionally P (D) is called the disc algebra and denoted by A(D) while P (S) is called the ball algebra and denoted by A(B). Note that by our definition a representing measure for A(D) will be defined on D, while a representing measure for A(B) will be defined on S. The motivation is that in this way we obtain an interesting set of representing measures for A(D), all of which can be “lifted”, to furnish representing measures on the sphere. From now on Φ will be point evaluation at a point a, usually 0, of the unit ball or disc and we will write MaS, respectively MaD, or when no confusion is possible Ma, instead of MΦ. 1991 Mathematics Subject Classification: 30H05, 32E25, 46J10, 46J15.