Vol. 78, No. 1, 1978

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Multiplicative linear functionals of Stein algebras

Robert Martin Ephraim

Vol. 78 (1978), No. 1, 89–93

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 = 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.

Mathematical Subject Classification 2000
Primary: 32E25
Secondary: 46J15
Received: 13 September 1977
Revised: 13 February 1978
Published: 1 September 1978
Robert Martin Ephraim