2020/09/24 by Xianzhe Feng, Denny H. Leung, Feng, Xianzhe +1
Mathematics · #46E15 #46E40 #47B38 #47H30 #53E35 #Algebraic structures and combinatorial models #FOS: Mathematics #Functional Analysis (math.FA) #Homotopy and Cohomology in Algebraic Topology #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2009.11570
openalex publication_date 2020/09/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 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.