Vol. 36, No. 3, 1971

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On dominated extensions in linear subspaces of 𝒞C (X)

E. M. Alfsen and B. Hirsberg

Vol. 36 (1971), No. 3, 567–584

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.

Mathematical Subject Classification
Primary: 46.25
Secondary: 31.00
Received: 18 June 1970
Published: 1 March 1971
E. M. Alfsen
B. Hirsberg