Extension of phase-isometries between the unit spheres of complex L∞(Γ) spaces
Abstract
Let Γ be nonempty index set, and X, Y are complex L∞(Γ)-type spaces. f: SX, SY will denote their unit spheres. Give a surjective mapping f: SX → SY satisfying the functional equation
{||f(x) + f(y)||, ||f(x)− f(y)||} = {||x+y||, ||x−y||} (x, y ∈ SX )
We show that there exists a function ε: SX → {−1,1} such that ε f is an isometry. Moreover, this isometry is the restriction of a real linear isometry from X to Y.Copyright © 2021 Yarong Zhang, Lingen Zhu, Jianan Yang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Cite this article
Y. Zhang, L. Zhu, J. Yang, Extension of phase-isometries between the unit spheres of complex L∞(Γ) spaces, Adv. Fixed Point Theory 11 (2021), 7. https://doi.org/10.28919/afpt/5323
BibTeX
@article{Zhang2021,
author = {Zhang, Yarong and Zhu, Lingen and Yang, Jianan},
title = {Extension of phase-isometries between the unit spheres of complex L∞(Γ) spaces},
journal = {Adv. Fixed Point Theory},
volume = {11},
year = {2021},
pages = {7},
doi = {10.28919/afpt/5323}
}