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

Abstract

Suppose that f : S_X → S_Y 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 : S_X → {−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.

Mathematics Subject Classification (2010):

• Primary: 46B03
• Secondary: 46B04

Key words:

• ^∞(Γ)-type spaces
• Wigner’s theorem
• phase equivalent
• Tingley’s problem