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

Abstract

Suppose that f : SX → SY is a surjective map between the unit spheres of two real ∞ (Γ)-type spaces X and Y satisfying the following equation 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 ∞ (Γ)-type spaces.