Yuanxia Li, Dongni Tan. Source: Annals of Functional Analysis, Volume 10, Number 4, 515--524.Abstract:
We say that a map $f:X\rightarrow Y$ between two real normed spaces is a phase-isometry if $\{\|f(x)+f(y)\|,\|f(x)-f(y)\|\}=\{\|x+y\|,\|x-y\|\}$ holds for all $x,y\in X$ . Two maps $f,g:X\rightarrow Y$ are called phase-equivalent if there is a phase function $\varepsilon :X\rightarrow \{-1,1\}$ such that $\varepsilon f=g$ . By studying the properties of surjective phase-isometries on the Tsirelson space $T$ , we show that such maps are phase-equivalent to linear isometries. This gives a real version of Wigner’s theorem for the Tsirelson space.