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

Elói Medina Galego ¹   ; Michael A. Rincón-Villamizar ²

¹ Department of Mathematics, IME University of São Paulo Rua do Matão 1010 São Paulo, Brazil
² Escuela de Matemáticas Facultad de Ciencias Universidad Industrial de Santander Carrera 27, calle 9 Bucaramanga, Colombia

Fundamenta Mathematicae, Tome 239 (2017) no. 2, pp. 185-200

Résumé

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, $1 \leq p \lt \infty$.

Détail

DOI : 10.4064/fm294-1-2017

Keywords: locally compact hausdorff space real banach lattice denote banach lattice x valued continuous functions vanishing infinity endowed supremum norm refine banach space results due cambern setting banach lattices prove there banach lattice isomorphism satisfying lambda homeomorphic where lambda inf max x y text geq result optimal classical spaces leq infty