Let (X,𝒪X) be a Stein analytic
space, and let 𝒪(X) denote the space of global sections of 𝒪X endowed
with its usual Frechet topology. The question of the continuity of complex
valued multiplicative linear functionals of 𝒪(X) will be studied. The main
result can be stated as follows: Theorem: Let (X,𝒪X) be a Stein space,
and let α;𝒪(X) →C be a multiplicative linear functional. Suppose one
can find an analytic subset Y ⊂ X such that all the connected components
of both Y and X − Y are finite dimensional. Then α must be continuous.
More generally, suppose that one can find a sequence of analytic subsets of
X, X = Y0⊃ Y1⊃⋯⊃ Yn= ∅, such that for any i,0 ≦ i < n, all the
connected components of Yi− Yi+1 are finite dimensional. Then α must be
continuous.
