A variant of Tingley's problem on ordered Banach spaces of absolutely continuous functions

Abstract

For each $1\le p\le\infty$ and $j=1,2$, let $AC^p(Ω_j)$ denote the Banach space of complex-valued absolutely continuous functions on a closed unit interval $Ω_j=[x_j,x_j+1]$. We equip $AC^p(Ω_j)$ with the $p$--norm $\|f\|_{AC,p}$, and the order $\ge_{AC}$ defined by $f(x_j)\ge0$ and $f'\ge 0$ a.e. Set $$S(AC^p(Ω_j))^+=\{f\in AC^p(Ω_j):\|f\|_{AC,p}=1,\ f\ge_{AC}0\}.$$ We prove that, for each $1\le p\le\infty$, every surjective isometry $S(AC^p(Ω_1))^+\to S(AC^p(Ω_2))^+$ extends uniquely to a complex--linear isometric order isomorphism from $AC^p(Ω_1)$ onto $AC^p(Ω_2)$. As an application, we obtain a corresponding extension theorem for surjective phase--isometries.