Two of the most important
global properties of complex spaces (X,𝒪), holomorphic convexity and holomorphic
separability, can each be characterized in terms of the standard natural map
χ : X → Sc(𝒪(X)), x → χx, χx(f) := f(x), f ∈𝒪(X), of X into the continuous
spectrum Sc(𝒪(X)) of the global function algebra 𝒪(X). The question as to whether
there is any global function theoretical property of (X,𝒪) corresponding to the
surjectivity of χ has remained unanswered. The purpose of this paper is to present an
answer for finite dimensional spaces. For such spaces (X,𝒪) it will be shown
that the surjectivity of χ is equivalent to requiring that for finitely many
functions f1,⋯,fm∈𝒪(X) with no common zero on X there exist functions
g1,⋯,gm∈𝒪(X) with ∑i=1mfigi= 1. This property will be called the weak
Nullstellensatz for the complex space (X,𝒪). An example due to H. Rossi
shows that this result is not valid for infinite dimensional complex spaces. An
application of the weak Nullstellensatz for Fréchet algebras A involving the
Michael conjecture is that Sc(A) is always dense in the spectrum S(A) of
A.
