The main result is the
following: Given a closed linear subspace A of 𝒞C(X) where X is compact Hausdorff
and A contains constants and separates points, and let F. be a compact subset of the
Choquet boundary ∂AX with the property that the restriction to F of every
A-orthogonal boundary measure remains orthogonal. If a0∈ A|F and a0≦ Ψ|F for
some strictly positive A-superharmonic function Ψ, then a0 can be extended to a
function a ∈ A such that a ≦ Ψ on all of X. It is shown how this result is related to
various known dominated extension-and peak set-theorems for linear spaces and
algebras. In particular, it is shown how it generalizes the Bishop-Rudin-Carleson
Theorem.
