Let (X,𝒪_{X}) be a Stein analytic
space, and let 𝒪(X) denote the space of global sections of 𝒪_{X} 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,𝒪_{X}) 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 = Y _{0} ⊃ Y _{1} ⊃⋯ ⊃ Y _{n} = ∅, such that for any i, 0 ≦ i < n, all the
connected components of Y _{i} − Y _{i+1} are finite dimensional. Then α must be
continuous.
