Banach-lattice isomorphisms of $C_0(K,X)$ spaces which determine the locally compact spaces $K$
Volume 239 / 2017
Fundamenta Mathematicae 239 (2017), 185-200
MSC: Primary 46B03, 46E15; Secondary 46E40, 46B25.
DOI: 10.4064/fm294-1-2017
Published online: 5 May 2017
Abstract
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, $1 \leq p \lt \infty$.
