Wigner’s theorem on the Tsirelson space T

Abstract

We say that a map $f:X\to Y$ between two real normed spaces is a phase-isometry if $\{\Vert f(x)+f(y)\Vert ,\Vert f(x)-f(y)\Vert \}=\{\Vert x+y\Vert ,\Vert x-y\Vert \}$ holds for all $x,y\in X$. Two maps $f,g:X\to Y$ are called phase-equivalent if there is a phase function $\epsilon :X\to \{-1,1\}$ such that $\epsilon f=g$. 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.