Rigidity of isometries on vector and normed lattices

Abstract

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 $L_1[0,1]$.