Phase-isometries on the unit sphere of CL-spaces

Abstract

A mapping f:SX→SY between the unit spheres of two real Banach spaces X and Y is said to be a phase-isometry if it satisfies{‖f(x)+f(y)‖,‖f(x)−f(y)‖}={‖x+y‖,‖x−y‖} for all x,y∈SX. When f is surjective, X is a real CL-space and Y is an arbitrary real Banach space, we establish in this paper that there exists a phase-function ε:SX→{−1,1} such that ε⋅f is an isometry which is the restriction of a linear isometry from X to Y.