On phase-isometries between the positive cones of continuous function spaces

Let K be a compact Hausdorff perfectly normal space and T be a compact Hausdorff space, C+(K) = {f is an element of C(K) : f(k) >= 0 for all k is an element of K} be the positive cone of C(K). In this paper, we show that if F : C+(K) -> C+(T) is a phase-isometry, that is {?F(f) + F(g)?, ?F(f) - F(g)?} = {?f + g?, ?f - g?}, for all f, g is an element of C+(K), then there exists a nonempty closed subset S subset of T , such that F(.)IS : C+(K) -> C+(S) (restriction of F(.) to S ) is an additive isometry (the restriction of a linear isometry between C(K) and C(S)). Moreover, if F is almost surjective, then K and T are homeomorphic and F is the restriction of a surjective linear isometry between C(K) and C(T) induced by the homeomorphism.