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 ℱ of the space C(U) of all
complex-valued continuous functions on the closure U of the set U. Let each
function in ℱ be A-holomorphic in U and assume that some but not all of
the functions in ℱ have zeros in U. Then the second main result is that
ℱ 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.
