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Banach-lattice isomorphisms of $C_0(K,X)$ spaces which determine the locally compact spaces $K$

Volume 239 / 2017

Elói Medina Galego, Michael A. Rincón-Villamizar Fundamenta Mathematicae 239 (2017), 185-200 MSC: Primary 46B03, 46E15; Secondary 46E40, 46B25. DOI: 10.4064/fm294-1-2017 Published online: 5 May 2017


For a locally compact Hausdorff space $K$ and a real Banach-lattice $X$ let $C_0(K, X)$ denote the Banach lattice of all $X$-valued continuous functions vanishing at infinity, endowed with the supremum norm.

We refine some Banach space results due to Cambern to the setting of Banach lattices to prove that if there is a Banach-lattice isomorphism $T$ from $C_0(K,X)$ onto $C_0(S,X)$ satisfying $$ \|T\|\, \|T^{-1}\| \lt \lambda^{+}(X), $$ then $K$ and $S$ are homeomorphic, where $$ \lambda^{+}(X)=\inf\{\max\{\|x-y\|,\|x+y\|\}:\|x\|= \|y\|= 1\text{ and }x,y\geq0\}. $$ This result is optimal for the classical $l_{p}$ spaces, $1 \leq p \lt \infty$.


  • Elói Medina GalegoDepartment of Mathematics, IME
    University of São Paulo
    Rua do Matão 1010
    São Paulo, Brazil
  • Michael A. Rincón-VillamizarEscuela de Matemáticas
    Facultad de Ciencias
    Universidad Industrial de Santander
    Carrera 27, calle 9
    Bucaramanga, Colombia

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