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 $A{C}^p({\Omega }_j)$ denote the Banach space of complex-valued absolutely continuous functions on a closed unit interval ${\Omega }_j=[x_j,x_j+1]$. We equip $A{C}^p({\Omega }_j)$ with the $p$--norm $\Vert f{\Vert }_{AC,p}$, and the order ${\ge }_{AC}$ defined by $f(x_j)\ge 0$ and $f^{\prime }\ge 0$ a.e. Set $$S(A{C}^p({\Omega }_j){)}^{+}=\{f\in A{C}^p({\Omega }_j):\Vert f{\Vert }_{AC,p}=1,f{\ge }_{AC}0\}.$$ We prove that, for each $1\le p\le \infty$, every surjective isometry $S(A{C}^p({\Omega }_1){)}^{+}\to S(A{C}^p({\Omega }_2){)}^{+}$ extends uniquely to a complex--linear isometric order isomorphism from $A{C}^p({\Omega }_1)$ onto $A{C}^p({\Omega }_2)$. As an application, we obtain a corresponding extension theorem for surjective phase--isometries.