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 C0(X)C0(X) and C0(Y)C0(Y), respectively. Suppose that for each triple of distinct points x,x′,x″∈Xx,x′,x″∈X, there exists f∈Af∈A such that |f(x)|≠|f(x′)||f(x)|≠|f(x′)| and f(x″)=0f(x″)=0. Also suppose that for each pair of distinct points y,y′∈Yy,y′∈Y, there exists g∈Bg∈B such that |g(y)|≠|g(y′)||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 Tf=ω(f∘φ)Tf=ω(f∘φ) on E   and Tf=ω(f∘φ¯) on ChB∖E for all f∈Af∈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.