Vol. 18, No. 2, 1966

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Analytic phenomena in general function algebras

C. E. Rickart

Vol. 18 (1966), No. 2, 361–377

Let Σ be a locally compact Hausdorff space and A an algebra of complex-valued continuous functions on Σ which contains the constant functions. Assume that Σ carries the weakest topology in which every function in A is continuous and that each homomorphism of A onto the complex numbers is given by evaluation at a point of Σ. Then A is called a natural algebra of functions on Σ. The motivating example for most of this paper is the algebra 𝒫 of all polynomials in n complex variables. It is readily verified that 𝒫 is in fact a natural algebra of functions on the n-dimensional complex space Cn. In the general setting an abstract analytic function theory is constructed for Σ with the natural algebra A playing a role analogous to that of 𝒫 in the case of Cn. For example, the concepts of A-holomorphic functions and, in terms of these functions, A-analytic varieties in Σ are introduced. The first main result obtained is that every A-analytic subvariety of a compact A-convex subset of Σ is itself A-convex. Next let Ω be any compact A-convex subset Σ and U a relatively open subset of Ω disjoint from the Šilov boundary of Ω with respect to A. Consider a connected subset of the space C(U) of all complex-valued continuous functions on the closure U of the set U. Let each function in be A-holomorphic in U and assume that some but not all of the functions in have zeros in U. Then the second main result is that must contain a function with zeros on the topological boundary of U relative to the space Ω. This implies a local property of A-convex hulls which generalizes an important result due to K. Oka for polynomially convex hulls in Cn.

Mathematical Subject Classification
Primary: 46.55
Secondary: 32.49
Received: 15 February 1965
Published: 1 August 1966
C. E. Rickart