Unbounded Disjointness Preserving Linear Functionals and Operators

TL;DR

This paper investigates unbounded disjointness preserving operators between Banach lattices, showing their decomposition into bounded and unbounded parts, with specific results for continuous functions on compact spaces.

Contribution

It introduces a decomposition of unbounded disjointness preserving operators into bounded and unbounded components, extending understanding of their structure in Banach lattices.

Findings

Disjointness preserving functionals separate points in infinite dimensional Banach lattices.

Every disjointness preserving operator is bounded on an order dense ideal.

Operators on $C(X)$) decompose into bounded and unbounded parts with specific properties.

Abstract

Let $E$ and $F$ be Banach lattices. We show first that the disjointness preserving linear functionals separate the points of any infinite dimensional Banach lattice $E$, which shows that in this case the unbounded disjointness operators from $E\to F$ separate the points of $E$. Then we show that every disjointness preserving operator $T:E\to F$ is norm bounded on an order dense ideal. In case $E$ has order continuous norm, this implies that that every unbounded disjointness preserving map $T:E\to F$ has a unique decomposition $T=R+S$, where $R$ is a bounded disjointness preserving operator and $S$ is an unbounded disjointness preserving operator, which is zero on a norm dense ideal. For the case that $E=C(X)$, with $X$ a compact Hausdorff space, we show that every disjointness preserving operator $T:C(X)\to F$ is norm bounded on an norm dense sublattice algebra of $C(X)$, which leads then to a decomposition of $T$ into a bounded disjointness operator and a finite sum of unbounded disjointness preserving operators, which are zero on order dense ideals.