向量格与赋范格上等距映射的刚性
Rigidity of isometries on vector and normed lattices
AI总结：
该研究探讨向量格与赋范格上等距映射的刚性，推广Baker定理，证明模等距、正范数等距的仿射性及相关性质，还证得非线性序-等距刚性定理，明确其不强制线性性。
AI中文摘要：
我们研究由等距、序论及模保持条件导出的向量格与赋范格上映射的刚性现象。首先探讨严格凸性在Baker非满射形式Mazur-Ulam定理中的作用，在此过程中发展中点单射函数理论，并研究其与凸性及单调性的联系。利用该框架，我们证明Baker定理的推广。作为首个主要应用，我们证明每个模等距（即满足对所有x,y∈X有|T(x)-T(y)|=|x-y|的映射T:X→Y，其中X是向量格Y的子格）是仿射映射，且具有保不交性的线性部分；当X是理想时，其线性部分还是X到自身的双射且为对合。在赋范格情形，我们还证明每个映入严格凸赋范格的正范数等距不仅是仿射映射，其线性部分还是格同态。最后，我们证明非线性序-等距刚性定理：经原点平移后，值域具有严格单调范数的赋范格间每个保序范数等距，保持上确界与下确界，且既保不交性又不交可加性。明确的L₁[0,1]上非线性等距族表明，这些结论不强制线性性。
英文摘要：
We study rigidity phenomena for maps on vector and normed lattices arising from isometric, order-theoretic, and modulus-preserving conditions. We first investigate the role of strict convexity in Baker's nonsurjective version of the Mazur-Ulam theorem. In the process, we develop a theory of midpoint injective functions and investigate their connections with convexity and monotonicity. Using this framework, we prove a generalization of Baker's theorem. As our first main application, we show that every modulus isometry, i.e., a map $T:X\to Y$, where $X$ is a sublattice of a vector lattice $Y$, satisfying $$|T(x)-T(y)|=|x-y|, \qquad x,y\in X,$$ is affine and has a disjointness-preserving linear part. When $X$ is an ideal, the linear part is moreover a bijection of $X$ onto itself and an involution. In the normed lattice setting, we also show that every positive norm isometry into a strictly convex normed lattice is not only affine, but has a linear part that is a lattice homomorphism. Finally, we prove a nonlinear order-isometric rigidity theorem. After shifting to the origin, every order-preserving norm isometry between normed lattices whose codomain has a strictly monotone norm, preserves suprema and infima and is both disjointness preserving and disjointly additive. These conclusions do not force linearity, as shown by an explicit family of nonlinear isometries on $L_1[0,1]$.
