Holomorphically Convex Sets in Several Complex Variables

Abstract

In this paper we study holomorphic functions from the point of view of function algebras. By a function algebra we mean an algebra A of continuous (complex-valued) functions on a compact Hausdorff space X which is closed in the sup norm, 11 * Ix. We assume also that A contains the constants and separates the points of X. In this case, it is well-known, the space of maximal ideals S(A) of A is a compact Hausdorff space in the weak topology determined by A; A can be represented as a closed algebra of continuous functions on S(A), and X can be embedded as a closed subset of S(A) [11]. Further, for any f e A, I I f I x = I I f I 1s(,). Any subset B of S(A) such that for all f eA, Iif lB = lIfIIS(A) is called a boundary for A. The intersection of all closed boundaries is a boundary, and is called the Silov boundary, denoted by 1(A) [12]. The Silov boundary is not necessarily the smallest boundary for A; the latter need not even exist. But if A is separable (as a Banach space), then S(A) is metric, and this minimal boundary does exist. In this case 1(A) is just the closure of the minimal boundary [4]. Let M be a complex analytic manifold, and let K be a compact subset of M. Let H(K) be the algebra of all functions holomorphic in a neighborhood of K, and let A(K) represent the closure of H(K) in the norm II IlK. How can we determine S(A(K)), 1(A(K)), and, as it makes sense in this case, since A(K) is separable, the minimal boundary? These questions have been discussed for certain types of compact sets in Cn (complex n-dimensional vector space) by K. de Leeuw [10], and K. Hoffman [14]. The papers of S. Bergman on distinguished boundary domains [3] are the first to indicate the significance of the Silov boundary in several complex variables (see also D. Lowdenslager [17] and H. Bremermann [8]). The Silov boundary is the smallest subset of K on which we can hope to represent holomorphic functions by an integral formula. If M is of dimension one (i.e., a Riemann surface), then K is the space of maximal ideals, and OK is the Silov boundary of A(K) [1, 4, 22]. In higher dimensions, neither of these is in any sense generally true. For example, if 470