$2$-Local Isometries on Spaces of Lipschitz Functions

Canadian mathematical bulletin

A. Jiménez-Vargas¹  Moisés Villegas-Vallecillos¹ 

[1] Departamento de Álgebra y Análisis Matemático, Universidad de Almería, 04120 Almería, Spain

关键词: isometry;    local isometry;    Lipschitz function;   

DOI  :  10.4153/CMB-2011-025-5

学科分类：数学（综合）

来源: University of Toronto Press * Journals Division

【 摘 要 】

Let $(X,d)$ be a metric space, and let $mathop{extrm{Lip}}(X)$ denote the Banachspace of all scalar-valued bounded Lipschitz functions $f$ on $X$endowed with one of the natural norms $| f| =max {| f| _infty ,L(f)}$ or $|f| =|f| _infty +L(f),$ where $L(f)$ is theLipschitz constant of $f.$ It is said that the isometrygroup of $mathop{extrm{Lip}}(X)$ is canonical if every surjective linear isometry of $mathop{extrm{Lip}}(X) $ is induced by a surjective isometry of $X$. In this paperwe prove that if $X$ is bounded separable and the isometry group of$mathop{extrm{Lip}}(X)$ is canonical, then every $2$-local isometryof $mathop{extrm{Lip}}(X)$ isa surjective linear isometry. Furthermore, we give a completedescription of all $2$-local isometries of $mathop{extrm{Lip}}(X)$ when $X$ isbounded.

【 授权许可】

Unknown