HOLOMORPHICALLY CONVEX SETS IN SEVERAL COMPLEX VARIABLES
HOLOMORPHICALLY CONVEX SETS IN SEVERAL COMPLEX VARIABLES
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多复变量的全纯凸集
DOI：
10.2307/1970292
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发表时间：
1961
影响因子：
4.9
通讯作者：
H. Rossi
中科院分区：
文献类型：
--
作者：
H. Rossi
本文从函数代数的角度研究了全纯函数。我们所说的函数代数是指在紧化Hausdorff空间X上的连续（复值）函数的代数a，该空间在sup范数11 * Ix中是封闭的。在这种情况下，众所周知，A的极大理想空间S(A)是由A确定的弱拓扑中的紧化Hausdorff空间；A可以表示为S(A)上连续函数的封闭代数，X可以嵌入为S(A)[11]的封闭子集。更进一步，对于任意f (A) I (x) = I (1) I （1,）S(A)的任何子集B满足对所有f eA， Iif lB = lIfIIS(A)称为A的边界。所有封闭边界的交集是一个边界，称为Silov边界，用1(A)[12]表示。Silov边界不一定是A的最小边界；后者甚至不必存在。但是如果A是可分离的（作为巴拿赫空间），那么S(A)是度规的，并且这个最小边界确实存在。在这种情况下，1(A)就是最小边界[4]的闭包。设M是复解析流形，设K是M的紧子集，设H(K)是K的邻域内所有全纯函数的代数，设a (K)表示H(K)在范数IlK中的闭包。我们如何确定S(A(K)), 1(A(K))，以及，在这种情况下，由于A(K)是可分离的，最小边界是有意义的？这些问题已经由K. de Leeuw[10]和K. Hoffman[14]讨论了Cn（复n维向量空间）中某些类型的紧集。S. Bergman关于区分边界域[3]的论文首次指出了Silov边界在几个复杂变量中的重要性（另见D. Lowdenslager[17]和H. Bremermann[8]）。Silov边界是K的最小子集，我们希望在其上用积分公式表示全纯函数。若M为1维（即黎曼曲面），则K为极大理想空间，OK为a (K)[1,4,22]的Silov边界。在更高的维度中，这两种情况在任何意义上都是不正确的。例如，如果470
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