# Phase-isometries on real normed spaces

## TL;DR

This paper investigates phase-isometries between real normed spaces, extending Wigner's theorem, and proves that surjective phase-isometries are essentially linear isometries multiplied by a sign, in specific classes of spaces.

## Contribution

It establishes that surjective phase-isometries are linear isometries times a sign function for smooth, 1(1) and 1(1) spaces, extending Wigner's theorem to these spaces.

## Key findings

- Surjective phase-isometries are linear isometries times a sign in smooth spaces.
- The result holds for 1(1) and 1(1) spaces.
- Extension of Wigner's theorem to real normed spaces.

## Abstract

We say that a mapping $f: X \rightarrow Y$ between two real normed spaces is a phase-isometry if it satisfies the functional equation \begin{eqnarray*} \{\|f(x)+f(y)\|, \|f(x)-f(y)\|\}=\{\|x+y\|, \|x-y\|\} \quad (x,y\in X).\end{eqnarray*} A generalized Mazur-Ulam question is whether every surjective phase-isometry is a multiplication of a linear isometry and a map with range $\{-1, 1\}$. This assertion is also an extension of a fundamental statement in the mathematical description of quantum mechanics, Wigner's theorem to real normed spaces. In this paper, we show that for every space $Y$ the problem is solved in positive way if $X$ is a smooth normed space, an $\mathcal{L}^{\infty}(\Gamma)$-type space or an $\ell^1(\Gamma)$-space with $\Gamma$ being an index set.