Nonlinear biseparating maps

TL;DR

This paper introduces a new framework for understanding nonlinear biseparating maps between vector-valued function spaces, characterizing their structure and continuity properties under mild assumptions.

Contribution

It proposes a revised definition of biseparating maps for nonlinear operators and characterizes these maps on various function spaces, extending previous additive results.

Findings

Biseparating maps are locally determined under mild conditions.

Characterizations of biseparating maps on continuous, uniformly continuous, and Lipschitz function spaces.

Automatic continuity results for certain classes of biseparating maps.

Abstract

An additive map $T$ acting between spaces of vector-valued functions is said to be biseparating if $T$ is a bijection so that $f$ and $g$ are disjoint if and only if $Tf$ and $Tg$ are disjoint. Note that an additive bijection retains $\mathbb{Q}$-linearity. For a general nonlinear map $T$, the definition of biseparating given above turns out to be too weak to determine the structure of $T$. In this paper, we propose a revised definition of biseparating maps for general nonlinear operators acting between spaces of vector-valued functions, which coincides with the previous definition for additive maps. Under some mild assumptions on the function spaces involved, it turns out that a map is biseparating if and only if it is locally determined. We then delve deeply into some specific function spaces -- spaces of continuous functions, uniformly continuous functions and Lipschitz functions -- and characterize the biseparating maps acting on them. As a by-product, certain forms of automatic continuity are obtained. We also prove some finer properties of biseparating maps in the cases of uniformly continuous and Lipschitz functions.