On phase-isometries between the positive cones of c0

Let c(0)(+) be the positive cone of c(0), i.e., c(0)(+) = {x = (x(n))(infinity)(n=1) is an element of c(0) : x(n) >= 0, for all n is an element of N}. A map f : c(0)(+) -> c(0)(+) is called a phase-isometry provided {parallel to f(x) + f(y)parallel to, parallel to f(x) - f(y)parallel to} = {parallel to x + y parallel to, parallel to x - y parallel to} for all x, y is an element of c(0)(+). In this paper, we prove that every phase-isometry f : c(0)(+) -> c(0)(+) is actually an isometry. And there exists a bounded linear operator T : (span) over bar f (c(0)(+)) -> c(0) with parallel to T parallel to = 1 such that Tf = Id(c+0). Furthermore, if f is almost surjective, then f is an additive isometry as the restriction of a surjective linear isometry from c(0) onto itself.