The Banach lattice $C[0,1]$ is super $d$-rigid

Y. A. Abramovich ¹   ; A. K. Kitover ²

¹ Department of Mathematical Sciences IUPUI Indianapolis, IN 46202, U.S.A.
² Department of Mathematics Community College of Philadelphia 1700 Spring Garden Street Philadelphia, PA 19130, U.S.A.

Studia Mathematica, Tome 159 (2003) no. 3, pp. 337-355

Résumé

The following properties of $C[0,1]$ are proved here. Let $T:C[0,1] \to Y$ be a disjointness preserving bijection onto an arbitrary vector lattice $Y$. Then the inverse operator $T^{-1}$ is also disjointness preserving$,$ the operator $T$ is regular$,$ and the vector lattice $Y$ is order isomorphic to $C[0,1]$. In particular if $Y$ is a normed lattice$,$ then $T$ is also automatically norm continuous. A major step needed for proving these properties is provided by Theorem 3.1 asserting that $T$ satisfies some technical condition that is crucial in the study of operators preserving disjointness.

Détail

DOI : 10.4064/sm159-3-1

Keywords: following properties proved here disjointness preserving bijection arbitrary vector lattice inverse operator disjointness preserving operator regular vector lattice order isomorphic particular normed lattice automatically norm continuous major step needed proving these properties provided theorem asserting satisfies technical condition crucial study operators preserving disjointness