Real-linear isometries between subspaces of continuous functions

Let X and Y be locally compact Hausdorff spaces. Let A and B be complex-linear subspaces of C 0 ( X ) and C 0 ( Y ) , respectively. Suppose that for each triple of distinct points x , x ′ , x ″ ∈ X , there exists f ∈ A such that | f ( x ) | ≠ | f ( x ′ ) | and f ( x ″ ) = 0 . Also suppose that for each pair of distinct points y , y ′ ∈ Y , there exists g ∈ B such that | g ( y ) | ≠ | g ( y ′ ) | . For such A and B, we prove the following statement: If T is a real-linear isometry of A onto B, then there exist an open and closed subset E of Ch B, a homeomorphism φ of Ch B onto Ch A and a unimodular continuous function ω on Ch B such that T f = ω ( f ∘ φ ) on E and T f = ω ( f ∘ φ ¯ ) on Ch B ∖ E for all f ∈ A , where Ch A and Ch B are the Choquet boundaries for A and B, respectively. Moreover, we remark that the separation condition on A cannot be omitted in the above result.