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

  • Dongni Tan
  • Xiaolei Xiong
Keywords: L∞(Γ)-type spaces, Wigner’s theorem, phase equivalent, Tingley’s problem.


Suppose that f : SX → SY 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.


Journal Identifiers

eISSN: 1727-933X
print ISSN: 1607-3606