Vol. 45, No. 1, 1973

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A boundary for the algebras of bounded holomorphic functions

Dong S. Kim

Vol. 45 (1973), No. 1, 269–274

Let (X,A) be a ringed space and let D be a domain in X. Let B = B(D) = {f A(D);fD < ∞}. A minimal boundary for B is defined as a unique smallest subset of D such that every function in B attains its supremum near the set. The followings are shown: If X is locally compact, D is relatively compact, and B separates the points of D then there exists a minimal boundary. Under the same assumptions, the natural projection of the Silov boundary B into X is the minimal boundary. If A is a maximum modulus algebra and the set of frontier points for A is the minimal boundary, then any holomorphic function which is bounded near the minimal boundary must be bounded. Finally, if D is the spectrum of B (with the compact open topology), then the topological boundary of D is the set of frontier points for B.

Mathematical Subject Classification 2000
Primary: 32E25
Secondary: 46J10
Received: 10 November 1971
Published: 1 March 1973
Dong S. Kim