Let be a closed, point-separating sub-algebra of , where X is a locally compact Hausdorff space. Assume that X is the maximal ideal space of . If , the set is denoted by . After characterizing the points of the Choquet boundary as strong boundary points this equivalence is used to complete the discussion initiated in a previous paper proving the

Main Theorem: If is a surjective map with the property that for every pair of functions , then there is an onto homeomorphism and a signum function on X such that , for all and .