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 $AC<sup>p(Ω_j)$ denote the Banach space of complex-valued absolutely continuous functions on a closed unit interval $Ω<em>j=[x_j,x_j+1]$. We equip $AC<sup>p(Ω_j)$ with the $p$--norm $|f|</em>{AC,p}$, and the order $\ge_{AC}$ defined by $f(x_j)\ge0$ and $f&#39;\ge 0$ a.e. Set $$S(AC^{p(Ω_j))^+={f\in} AC^{p(Ω<em>j):|f|</em>{AC,p}=1,\} f\ge_{AC}0}.$$ We prove that, for each $1\le p\le\infty$, every surjective isometry $$S(AC^{p(Ω_1))^+\to} S(AC<sup>p(Ω_2))<sup>+$$ extends uniquely to a complex--linear isometric order isomorphism from $AC<sup>p(Ω_1)$ onto $AC<sup>p(Ω_2)$. As an application, we obtain a corresponding extension theorem for surjective phase--isometries.

• The paper establishes that every surjective isometry on the positive unit sphere of AC^p spaces uniquely extends to a full complex-linear isometric order isomorphism.
• It employs innovative geometric and order-theoretic methods that bypass traditional convex body approaches, particularly addressing challenges in the L^1 setting.
• The explicit extension formula and structural identifications provided pave the way for further studies on isometric and phase-isometric transformations in ordered Banach spaces.

A Variant of Tingley's Problem on Ordered Banach Spaces of Absolutely Continuous Functions

Introduction and Motivation

The paper addresses an order-theoretic variant of Tingley's problem in the context of Banach spaces of absolutely continuous functions, specifically focusing on the ordered Banach spaces $AC^p(\Omega)$, $1 \leq p \leq \infty$, where $\Omega$ is a closed unit interval. Tingley's problem inquires whether every surjective isometry between the unit spheres of Banach spaces extends to a surjective linear isometry between the underlying spaces. The modification considered in this work restricts attention to the positive part of the sphere, a natural consideration in the setting of ordered Banach spaces with generating cones.

Classical results affirm the extension property for $L^p$ spaces with $1\leq p\leq \infty$ for both spheres and their positive parts, but analogous results are subtle and historically unsettled for spaces of absolutely continuous functions. This paper establishes that for every $1 \leq p \leq \infty$, surjective isometries between the positive unit spheres of $AC^p(\Omega)$ do indeed extend uniquely to complex-linear isometric order isomorphisms of the entire spaces. The analysis leverages specific geometric and order-theoretic structures of $AC^p(\Omega)$, employing a novel mechanism distinct from the convex body approach previously successful for $C^1$ and $\operatorname{Lip}$ spaces.

Main Results

The central theorem asserts that for any $1 \leq p \leq \infty$0, every surjective isometry $1 \leq p \leq \infty$1 admits a unique extension to a complex-linear isometric order isomorphism $1 \leq p \leq \infty$2. The extension takes an explicit constructive form: $1 \leq p \leq \infty$3 for $1 \leq p \leq \infty$4, $1 \leq p \leq \infty$5, where $1 \leq p \leq \infty$6 is the unique complex-linear isometric order isomorphism derived from the behavior of $1 \leq p \leq \infty$7 on the positive spheres.

The arguments are structurally partitioned according to $1 \leq p \leq \infty$8:

• For $1 \leq p \leq \infty$9 and $\Omega$0, the proof relies on the order-preserving isometric identification of $\Omega$1 with a direct sum $\Omega$2. The analysis of the positive unit sphere reduces to properties of the $\Omega$3 positive unit ball, through explicit bijections between elements and their derivatives.
• For $\Omega$4, a special identification is invoked, realizing $\Omega$5 as an $\Omega$6 space over the disjoint union $\Omega$7, and employing structural arguments about complex-linear isometric order isomorphisms of $\Omega$8 spaces with atoms.

Crucially, the proof overcomes the failure of the "convex body" approach for $\Omega$9 spaces due to the positive unit ball structure in $L^p$0 lacking nonempty interior. Instead, it exploits careful metric separation properties and the extension theorem for positive isometries of $L^p$1 spaces.

Technical Analysis

The author establishes several foundational isometric order isomorphisms:

• $L^p$2, where $L^p$3.
• For $L^p$4, the space $L^p$5 is further realized as $L^p$6, with $L^p$7 a space with an isolated atom.

Key technical components include:

• Demonstration of explicit bijections between positive parts of the $L^p$8 sphere and the $L^p$9 positive unit ball.
• Geometric and metric separation lemmas characterizing how the derivative coordinate and norm behavior uniquely determine the isometric extension.
• Verification that the extension constructed from sphere isometries preserves both linear and order structures, leveraging recent classification results for $1\leq p\leq \infty$0 spaces [LNW21].

For $1\leq p\leq \infty$1, the paper introduces a decomposition of the positive unit sphere via sets $1\leq p\leq \infty$2 (fixed scalar coordinate) and $1\leq p\leq \infty$3 (fixed $1\leq p\leq \infty$4-norm coordinate), each requiring distinct isometric analysis.

Application to Phase-Isometries

An immediate corollary is that any surjective phase-isometry (an isometry respecting phase equivalence) between positive unit spheres of $1\leq p\leq \infty$5 spaces extends to a unique complex-linear isometric order isomorphism on the entire space. The same explicit formula for the extension applies, with the isomorphism $1\leq p\leq \infty$6 determined by the phase-isometry.

Implications and Future Directions

The paper closes a significant gap in the landscape of Tingley's problem for ordered function spaces. The explicit extension formula and uniqueness yield structural rigidity results for convex- and order-preserving transformations of positive spheres in these spaces. The methods circumvent the failure of prior convex geometric techniques for the $1\leq p\leq \infty$7 case, offering a template for further extensions to more general ordered Banach function spaces and possibly vector-valued or operator-valued absolutely continuous function spaces.

Given the robustness of the approach for all $1\leq p\leq \infty$8, it is expected that this work will guide the investigation of isometric and phase-isometric extension properties in Banach lattices and other function spaces with order structure. The metric and order-theoretic tools developed may also have implications for the classification of positive isometric embeddings in geometric analysis and functional representation theory.

Conclusion

This work establishes that, for ordered Banach spaces of absolutely continuous functions on compact intervals with $1\leq p\leq \infty$9, all surjective isometries (and phase-isometries) between the positive unit spheres uniquely extend to complex-linear isometric order isomorphisms of the spaces. The extension procedure is precisely characterized and exploits deep structural properties of these spaces, notably advancing the theory of isometries in function space settings and resolving the order-theoretic Tingley's problem in this context (2607.10685).