The Banach lattice $C[0,1]$ is super $d$-rigid
Volume 159 / 2003
Studia Mathematica 159 (2003), 337-355
MSC: Primary 47B60.
DOI: 10.4064/sm159-3-1
Abstract
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.
