WIGNER'S THEOREM IN L∞(Γ)-TYPE SPACES

We investigate surjective solutions of the functional equation {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} (x, y is an element of X), where f : X -> Y is a map between two real L-infinity(Gamma)-type spaces. We show that all such solutions are phase equivalent to real linear isometries. This can be considered as an extension of Wigner's theorem on symmetry for real L-infinity(Gamma)-type spaces.