Let (X,A) be a ringed space
and let D be a domain in X. Let B = B(D) = {f ∈ A(D);∥f∥D< ∞}. 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.
