A note on Tingley’s problem and Wigner’s theorem in the unit sphere of L^∞(Г)-type spaces

Abstract

Suppose that f : S_X → S_Y is a surjective map between the unit spheres of two real L^∞(Γ)-type spaces X and Y satisfying the following equation {∥f(x) + f(y)∥,∥f(x) − f(y)∥} = {∥x + y∥,∥x − y∥} (x,y ∈ SX). We show that such a mapping f is phase equivalent to an isometry, i.e., there exists a function ε : SX → {−1,1} such that εf is an isometry. We further show that this isometry is the restriction of a linear isometry between the whole spaces. These results can be seen as a combination of Tingley’s problem and Wigner’s theorem for L∞(Γ)-type spaces.