Vol. 99, No. 1, 1982

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The weak Nullstellensatz for finite-dimensional complex spaces

Sandra Hayes

Vol. 99 (1982), No. 1, 45–56

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.

Mathematical Subject Classification 2000
Primary: 32E25
Secondary: 46J15
Received: 11 June 1980
Published: 1 March 1982
Sandra Hayes