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$2$-Local Isometries on Spaces of Lipschitz Functions

  Published:2011-03-05
 Printed: Dec 2011
  • A. Jiménez-Vargas,
    Departamento de Álgebra y Análisis Matemático, Universidad de Almería, 04120 Almería, Spain
  • Moisés Villegas-Vallecillos,
    Departamento de Álgebra y Análisis Matemático, Universidad de Almería, 04120 Almería, Spain
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Abstract

Let $(X,d)$ be a metric space, and let $\mathop{\textrm{Lip}}(X)$ denote the Banach space 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 the Lipschitz constant of $f.$ It is said that the isometry group of $\mathop{\textrm{Lip}}(X)$ is canonical if every surjective linear isometry of $\mathop{\textrm{Lip}}(X) $ is induced by a surjective isometry of $X$. In this paper we prove that if $X$ is bounded separable and the isometry group of $\mathop{\textrm{Lip}}(X)$ is canonical, then every $2$-local isometry of $\mathop{\textrm{Lip}}(X)$ is a surjective linear isometry. Furthermore, we give a complete description of all $2$-local isometries of $\mathop{\textrm{Lip}}(X)$ when $X$ is bounded.
Keywords: isometry, local isometry, Lipschitz function isometry, local isometry, Lipschitz function
MSC Classifications: 46B04, 46J10, 46E15 show english descriptions Isometric theory of Banach spaces
Banach algebras of continuous functions, function algebras [See also 46E25]
Banach spaces of continuous, differentiable or analytic functions
46B04 - Isometric theory of Banach spaces
46J10 - Banach algebras of continuous functions, function algebras [See also 46E25]
46E15 - Banach spaces of continuous, differentiable or analytic functions
 

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